FROM  -THE-  LI  BRARY-  OF 
•WILLIAM -A  HILLEBRAND 


PHYSICS  OEPT. 


ELECTRIC  TRANSIENTS 


cMsQra&)'3/ill  Book  &  1m 

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ELECTRIC  TRANSIENTS 


BY 

CARL  EDWARD  MAGNUSSON 

AUTHOR  OF  "ALTERNATING  CURRENTS,"  PROFESSOR  OF  ELECTRICAL  ENGINEERING, 
DEAN  OF  THE  COLLEGE  OF  ENGINEERING,  DIRECTOR  OF  THE  ENGINEERING 

EXPERIMENT    STATION,    UNIVERSITY    OF    WASHINGTON 

A.  KALIN 

INSTRUCTOR    IN    ELECTRICAL    ENGINEERING,    UNIVERSITY    OF    WASHINGTON 

J.  R,  TOLMIE 

INSTRUCTOR    IN    ELECTRICAL    ENGINEERING,    UNIVERSITY    OF    WASHINGTON 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6&  8  BOUVERIE  ST.,  E.  C.  4 

1922 


COPYRIGHT,  1922,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


THE    MAPLE    PRESS     -     YORK    PA 


PREFACE 

Transient  electric  phenomena  generally  increase  in 
commercial  importance  with  the  size  and  complexity  of 
electric  systems,  and  a  knowledge  of  the  fundamental 
principles  of  electric  transients  and  their  application  to  the 
solution  of  quantitative  problems  is  as  essential  to  the 
successful  operation  of  large  power  and  communication 
systems  as  a  mastery  of  the  basic  laws  of  direct  and  alter- 
nating currents. 

This  work  is  an  outline  of  an  introductory  lecture  and 
laboratory  course  given  during  the  past  twelve  years  to 
electrical  engineering  students  in  the  University  of  Wash- 
ington. The  purpose  of  the  book  is  to  aid  the  student  in 
gaining  clear  concepts  of  the  fundamental  principles  of 
electric  transient  phenomena  and  their  application  to 
quantitative  problems.  The  course  as  outlined  is  pro- 
fessedly of  an  elementary  character  with  emphasis  placed 
on  the  physical  properties  of  electric  transients.  The  text 
is  illustrated  and  supplemented  by  a  large  number  of 
oscillograms  of  transients  that  occur  in  the  various  types  of 
machines  and  electric  circuits  in  common  use  in  electrical 
engineering  laboratories.  The  problems  are  based  on 
quantitative  data  obtained  from  laboratory  experiments 
under  circuit  conditions  that  may  easily  be  reproduced  by 
the  student. 

Quantitative  laboratory  work  is  essential  in  order  to 
readily  gain  insight  into  the  physical  nature  of  transient 
electric  phenomena.  It  is  advisable  to  require  the  student 
to  devote  at  least  two-thirds  of  the  time  allotted  to  a  course 
in  electric  transients  to  the  taking  of  oscillograms.  Adjust- 
ing an  oscillograph  so  as  to  obtain  sharply  defined,  well 
proportioned  oscillograms  of  electric  transients  is  an 
effective  method  for  acquiring  due  appreciation  of  quanti- 


995863 


vi  PREFACE 

tative  values,  both  absolute  and  relative,  of  the  factors 
involved.  The  quality  of  the  photographic  record  depends 
as  much  on  painstaking  care  in  handling  the  films  and  in 
developing  and  printing  the  oscillograms  as  on  skilful 
operation  of  the  oscillograph.  Many  pitfalls  in  the  photo- 
graphic part  of  the  work  may  be  avoided  by  carefully 
following  the  directions  given  in  the  Appendix. 

No  attempt  is  made  to  give  references  to  original  investi- 
gations or  to  papers  and  books  dealing  with  the  various 
phases  of  electric  transient  phenomena,  as  the  principles 
discussed  are  well  established  and  the  material  is  arranged 
in  text  book  form.  A  distinctive  feature  of  the  book  lies  in 
the  illustrations.  All  of  the  oscillograms  were  taken  by 
A.  Kalin  and  J.  R.  Tolmie  or  by  students  in  the  course 
under  their  direction  in  the  electrical  engineering 
laboratories  of  the  University  of  Washington. 

C.  EDWARD  MAGNUSSON. 

SEATTLE,  WASH., 
March,  1922. 


CONTENTS 

PAGE 

PREFACE v 

CHAPTER  I.— INTRODUCTION 1 

Magnetic  circuit — Dielectric  circuit — Electric  circuit. 

CHAPTER  II.— OSCILLOGRAPHS 10 

Three  element  oscillographs — Timing  wave  from  oscillator  genera- 
tor— Oscillograms — Problems  and  experiments. 

CHAPTER  III. — SINGLE  ENERGY  TRANSIENTS.     DIRECT  CURRENTS  .     23 
Single  energy  circuits — The  exponential  law — The  time  constant 
— Dissipation  or  attenuation  constants — The  exponential  curve — 
Initial    transient    values — Current,    voltage    and  ^magnetic   flux 
transients — Problems  and  experiments.        ^'4 

CHAPTER    IV. — SINGLE       ENERGY       TRANSIENTS.     ALTERNATING 

CURRENTS 40 

Single  phase,  single  energy  load  circuit  transients — Three  phase, 
single  energy  load  circuit  transients — Starting  transient  of  a  poly- 
phase rotating  magnetic  field — Polyphase  short  circuits.  Alter- 
nator armature  and  field  transients — -Single  phase  short  circuits. 
Alternator  armature  and  field  transients — Single  phase  short  cir- 
cuits on  polyphase  alternators — Problems  and  experiments. 

CHAPTER  V.— DOUBLE  ENERGY  TRANSIENTS 75 

Double  energy  circuits — Surge  or  natural  impedance  and  admit- 
tance— Frequency  of  oscillations  in  double  energy  circuits — 
Dissipation  constant  and  damping  factor  in  simple  double  energy 
circuits — Equations  for  current  and  voltage  transients — Problems 
and  experiments. 

CHAPTER  VI. — ELECTRIC  LINE  OSCILLATIONS.     SURGES  AND  TRAVEL- 
ING WAVES 101 

Artificial  transmission  lines — Time,  space  and  phase  angles — • 
Natural  period  of  oscillation — Length  of  line — Velocity  unit 
of  length — Surge  impedance — Voltage  and  current  oscillations  and 
power  surges — General  transmission  line  equations — Traveling 
waves — Compound  circuits — Problems  and  experiments. 

CHAPTER  VII.— VARIABLE  CIRCUIT  CONSTANTS 130 

Variable  resistance — Variable  inductance — Variable  conductance — 
Variable  condensance — Problems  and  experiments. 

vii 


viii  CONTENTS 

PAGE 

CHAPTER  VIII.— RESONANCE 145 

Voltage  resonance — Current  resonance — Coupled  circuits — Direct 
coupling — Inductive  coupling — Condensive  coupling — Coupling 
coefficient — Multiplex  resonance — Resonance  growth  and  decay — 
Problems  and  experiments. 

CHAPTER  IX.— OSCILLOGRAMS 168 

Starting  transients  of  a  D.C.  lifting  magnet — Opening  of  D.C.  and 
A.C.  circuit  breakers  due  to  overload — T.A.  regulator  operating 
transients — Short  circuits  on  series  generators — "Bucking 
broncho"  transients — Current  transformer  transients — Single 
phase  short  circuit  on  a  two  phase  alternator — Undamped  oscillo- 
graph vibrator  oscillations — Starting  transients  on  a  three  phase 
induction  motor — Starting  transients  on  a  repulsion-induction 
motor  and  a  split  phase  motor — Single  phase  operation  of  a  three 
phase  induction  motor — Transients  in  three  phase  induction  motor 
due  to  short  circuit  on  stator  terminals — Short  circuits  on  a  rotary 
converter — Synchronizing  a  rotary  converter  from  85  per  cent 
synchronous  speed — Synchronous  motor  falling  out  of  step  due  to 
overload — The  magnetic  flux  distribution  of  a  synchronous  motor 
when  slipping  a  pole — Problem  and  experiments. 

APPENDIX 189 

Instructions'  for  developing  and  printing  oscillograms. 
INDEX.  .    193 


ELECTRIC  TRANSIENTS 

CHAPTER  I 
INTRODUCTION 

The  laws  for  direct  currents,  as  usually  -expressed,  state 
the  relations  of  the  several  factors  involved  under  continu- 
ous or  permanent  conditions,  and  cannot  be  correicyy 
applied  while  the  current  or  voltage  is  increasing  or  decreas- 
ing. Similarly,  alternating  currents  are  expressed  as 
continuous  phenomena  by  means  of  effective  values  and 
complex  quantities,  on  the  basis  that  the  successive  cycles 
are  of  the  same  magnitude  and  wave  shape.  Observations 
and  test  data  for  both  the  direct-current  and  alternating- 
current  systems  are  ordinarily  taken  only  during  steady 
or  permanent  conditions.  The  equations  derived,  and  the 
data  obtained  from  tests,  apply  only  to  permanent  or 
constant  conditions  and  cannot  be  correctly  applied  during 
transition  periods  when  the  conditions  vary.  Transient 
electric  phenomena,  as  the  term  implies,  are  usually  of 
short  duration  and  relate  to  what  occurs  in  an  electric 
circuit  between  periods  of  stable  conditions.  This  defini- 
tion is,  however,  not  rigidly  adhered  to  in  electrical  discus- 
sions. Frequently  other  disturbances  that  militate  against 
successful  operation  of  electric  systems,  such  as  unstable 
electric  equilibrium,  permanent  instability,  resonance  and 
cumulative  oscillations  are  included  with  the  true  transients 
under  the  caption  of  transient  electric  phenomena. 

It  is  important  that  the  student  should  realize  that 
electric  transients  are  of  very  frequent  occurrence  in  all 
commercial  electric  systems.  Any  change,  such  as  the 
starting  or  stopping  of  a  motor,  the  turning  on  of  a  lamp, 
or  any  change  in  the  operating  conditions  necessitates  a 


2  ELECTRIC  TRANSIENTS 

re-adjustment  of  the  energy  content  in  the  whole  system 
and  produces  electric  transients  just  as  truly  as  a  stroke 
of  lightning  or  a  short  circuit.  In  the  operation  of  street 
car  systems  the  changes  in  load,  and  hence  the  transients 
on  the  system,  are  so  frequent  that  they  overlap  and  occupy 
by  far  the  greater  part  of  the  time;  hence,  for  street  railway 
systems,  it  might  appear  simpler  to  define  the  permanent  or 
steady  conditions  as  short  periods  occurring  between  succes- 
sive series  of  overlapping  transients. 

Electrical ^engineering  deals  with  the  transmission  and 
transformation  of  electric  energy.  During  permanent 
conditions- the  flow  of  energy  is  uniform  and  continuous; 
any  change  in  the  power  indicates  a  transient  condition. 
Changes  in  the  current  and  voltage  factors  imply  a  cor- 
responding change  in  the  energy  content  of  the  electric 
field,  since  a  magnetic  field  surrounds  all  electric  currents, 
and  an  increase  or  decrease  in  the  current  necessitates  a 
corresponding  change  in  the  stored  magnetic  energy. 
Similarly,  any  change  in  voltage  between  conductors  must 
be  accompanied  by  a  corresponding  re-adjustment  in  the 
energy  stored  in  the  dielectric  field  of  the  system. 

Magnetic  Circuit.— In  the  study  of  transient  phenomena, 
as  well  as  of  all  phases  of  the  electric  field,  Faraday's 
concept  of  magnetic  and  dielectric  lines  of  force  is  of  funda- 
mental importance.  All  magnetic  lines  are  continuous  and 
closed  on  themselves.  Ohm's  law  applies  to  the  magnetic 
circuits  in  the  same  way  as  to  the  electric  circuit.  The 
magnetic  flux  produced  is  equal  to  the  magneto-motive 
force  divided  by  the  reluctance. 

-,,         ,.    a  magneto-motive  force 

Magnetic  flux  =  -  : 

reluctance 

cy 

$  =  -    or  ff  =  (R$  (1) 

The  magnetic  field  is  produced  by,  and  is  proportional  to, 
the  electric  current. 

$  =  Li  (2) 


INTRODUCTION 


The  proportionality  factor  L  is  called  the  inductance  of 
the  circuit. 

The  reluctance  varies  directly  as  the  length  and  inversely 
as  the  cross  section  of  the  magnetic  circuit.  The  specific 
reluctance  per  cm.3  is  the  reciprocal  of  the  permeability  ju. 
If  the  magneto-motive  force  is  expressed  in  ampere  turns, 
the  resultant  field  intensity  is  given  by  the  equation. 

H  ----  4irnl  lO^per  cm.  (3) 

This  magnetizing  force  produces  a  magnetic  flux  density 
of  B  lines  per  cm.2  in  materials  having  /*  permeability. 

B  =  [J.H  lines  per  cm.2  (4) 

The  permeability  is  the  reciprocal  of  the  specific  reluct- 
ance in  the  magnetic  circuits  and  corresponds  to  the  specific 


FIG.   1. — Magnetic  field  of  single  conductor. 


Magnetic  field  of  circuit. 


conductivity  of  the  conductor  in  the  electric  circuits.  In 
empty  space  ^  =  1  and  for  all  non-magnetic  materials  it  is 
very  nearly  equal  to  unity.  For  magnetic  materials  the 
permeability  is  greatly  increased  and  may  reach  several 
thousand  for  soft  iron  and  steel.  The  factor  4?r  comes  from 
the  definition  of  a  unit  magnetic  pole  as  having  one  line 
per  cm.2  on  the  surface  of  a  sphere  of  unit  radius.  The 
10"1  factor  results  from  the  definition  of  the  ampere. 

In  building  up  a  magnetic  field,  lines  of  force  cut  the 
conductor  and  thus  produce  a  counter  e.m.f.,  or  inductance 


4  ELECTRIC  TRANSIENTS 

voltage,  Le,  which  is  equal  to  the  time  rate  of  change  of  the 
interlinked  magnetic  flux. 

d$       T  di 

*  -  dt  =  Ldt  <5> 

Necessarily  an  equal  opposite  voltage  must  be  impressed 
to  force  the  current  through  the  electric  circuit.  The  prod- 
uct of  the  voltage  and  the  current  represents  the  power 
required  to  generate  the  field.  Hence,  the  energy  stored  in 
a  magnetic  field  by  a  current,  7,  in  a  circuit  having  an  induc- 
tance, L,  is  given  by  equations  (6)  and  (7). 

Cw        C1  C1. 

I  dw  =   I  Leidt  =  L  I  idi  (6) 

Jo  Jo  Jo 

w-% 

The  energy  is  stored  magnetically  in  the  electric  field 
surrounding  the  conductor  and  is  proportional  to  the  square 
of  the  current.  When  the  current  decreases  the  energy  is 
returned  to  the  circuit,  for  if  i  and  therefore  <f>  decrease, 
di/dt  and  hence  Le  are  negative,  which  means  that  the  energy 
is  returned  to  the  electric  circuit. 

The  practical  unit  of  inductance,  L,  is  the  henry.  In 
any  consistent  system  of  units  a  circuit  possesses  one  unit 
of  inductance,  if  a  unit  rate  of  change  of  current  in  the 
circuit  generates  or  consumes  one  unit  of  voltage.  If  the 
current  changes  at  the  rate  of  one  ampere  per  second,  and 
the  voltage  generated  or  consumed  is  one  volt,  then  the 
inductance  is  one  henry. 

Dielectric  Circuit. — For  the  dielectric  field  similar  rela- 
tions exist.  All  dielectric  lines  of  force  are  continuous  and 
end  on  conductors.  Ohm's  Law  may  be  applied  to  the 
dielectric  circuit  in  the  same  manner  as  to  the  magnetic 
and  electric  circuits. 

Dielectric  flux  =  —, —  -— ^— -; 
elastance 

*  =      =  Ce  (8) 


INTRODUCTION 


The  dielectric  flux  is  directly  proportional  to  the  voltage 
between  the  conductors  and  inversely  proportional  to  the 
elastance  of  the  dielectric  circuit.  The  elastance,  S,  is  the 
reciprocal  of  the  condensance,  C,  and  varies  directly  as 
the  length,  x,  and  inversely  as  the  cross  section,  A,  of 
the  dielectric  circuit.  It  corresponds  to  resistance  of  the 
electric  circuit  and  to  reluctance  of  the  magnetic  circuit. 


FIG.  3. — Dielectric  field  of  single 
conductor. 


FIG.  4. — Dielectric  field  of  circuit. 


S  =       .  ;  C  =  -   '-  in  c.g.s.  electrostatic  units 
K.A  4:irX 


(9) 


S  =  -     —',  C  =  -.      -in  electromagnetic  units  (10) 

' 


i  ,         £ 

-1  -a;  darafs 

•,  A  1  O9  *  A 

C  =  -  =  88.42  *~10-15  farads 

X 


C  =  88.42  "     10-9  microfarads 

JU 


(12) 
(13) 


The  permittivity  K  is  unity  for  empty  space  and  very 
nearly  equal  to  unity  for  air  and  many  other  materials. 
In  Table  I  is  given  the  permittivity  constants  for  the  more 
common  dielectrics  used  in  electric  apparatus.  The  con- 
stant v  =  3-1010  cm/sec.,  the  velocity  of  the  propagation 
of  an  electric  field  in  space  (equivalent  to  the  velocity  of 
light)  ,  is  the  ratio  of  the  units  used  in  the  electromagnetic 


6 


ELECTRIC  TRANSIENTS 


and  electrostatic  systems.     The  factor  4w  comes  from  the 
definition  of  a  unit  line  of  dielectric  force. 

TABLE  I 


Material 

Permit- 
tivity 

Material 

Permit- 
tivity 

Air  and  other  gases  .... 

1.0 

Olive  oil  

3  .  0  to  3  2 

Alcohol,  amyl  
Alcohol,  ethyl  
Alcohol,  methyl  
Asphalt  
Bakelite  
Benzine  
Benzol 

15.0 
24.3  to  27.4 
32.7 
4.1 
6.6  to  16.0 
1.9 
2  2  to    24 

Paper  with  turpentine 
Paper  or  jute  impreg- 
nated   
Paraffin  
Paraffin  oil  
Petroleum  
Porcelain 

2.4 

4.3 
2.3 
1.9 
2.0 
5  3 

Condensite  
Glass  (easily  fusible)  .  . 
Glass  (difficult  to  fuse) 
Gutta-percha  
Ice  
Marble  

6.6  to  16.0 
2  .  0  to    5.0 
5.0  to  10.0 
3  .  0  to    5.0 
3.0 
6.0 

Rubber  
Rubber  vulcanized.  .  . 
Shellac  
Silk  
Sulphur  
Turpentine 

2.4 

2  .  5  to  3  .  5 
2.7  to  4.1 
1.6 
4.0 
2  2 

Mica  

5.0to    7.0 

Varnish  

2.0  to  4.1 

Micarta  

4.1 

The  charging  current,  ci,  storing  energy  in  the  dielectric 
circuit  is  equal  to  the  time  rate  of  change  in  the  dielectric 
flux. 

•  —  ^  —  r<^e  n/n 

=  dT     Ldt 

Hence  the  energy  stored  in  the  dielectric  field  by  a  voltage, 
E,  in  a  circuit  having  a  condensance,  C,  is  given  by  equa- 
tions (15)  and  (16): 


JW  (*E  /» 

dw  =   I  ciedt  =  C  I 


E 

ede 


CE* 
2 


(15) 
(16) 


The  energy  stored  dielectrically  in  the  electric  field  sur- 
rounding a  conductor  is  proportional  to  the  square  of  the 
voltage.  When  the  voltage  decreases  the  energy  is 
returned  to  the  electric  circuit,  for  if  e  and  therefore  ^ 


INTRODUCTION  7 

decreases,  then  de/dt  and  hence  ci  are  negative,  which 
means  that  the  energy  is  returned  to  the  electric  circuit. 
The  unit  of  condensance  (capacitance),  C,  is  the  farad. 
In  any  consistent  system  of  units  a  circuit  possesses  one 
unit  of  condensance  if  a  unit  rate  of  change  of  voltage 
produces  (or  consumes)  one  unit  of  current.  If  the  voltage 
changes  at  the  rate  of  one  volt  per  second  and  the  current 
produced  (or  consumed)  is  one  ampere,  the  condensance 


FIG.  5. — Electric  field  of  conductor. 


FIG.  C. — Electric  field  of  circuit. 


of  the  circuit  is  one  farad.  The  farad  is  too  large  a  unit  for 
practical  purposes  and  hence  in  commercial  problems  the 
condensance  is  usually  measured  in  microfarads. 

1  farad  =  106  microfarads  (17) 

Electric  Circuit. — The  electric  circuit  relates  specifically 
to  the  conductor  carrying  the  electric  current  although  the 
term  is  frequently  made  to  include  the  dielectric  and  mag- 
netic fields,  since  the  electric,  dielectric  and  magnetic 
circuits  are  interlinked.  Under  steady  or  permanent 
conditions  in  a  direct  current  system  the  electric  circuit 
transmits  the  energy  without  causing  any  change  in  the 
energy  stored  magnetically  and  dielectrically  in  the  space 
surrounding  the  electric  circuit.  In  starting  the  system  a 
transient  condition  exists  until  the  magnetic  and  dielectric 
fields  have  been  supplied  with  the  required  amount  of 
energy  as  determined  by  the  magnitude  of  the  current  and 
voltage  and  the  circuit  constants. 


8  ELECTRIC  TRANSIENTS 

If  the  electric  circuit  be  considered  as  something  separate 
and  apart  from  the  surrounding  magnetic  and  dielectric 
fields,  no  storage  of  energy  would  be  involved  and  hence 
no  transients  could  exist,  since  all  the  changes  would  be 
instantaneous.  But  the  electric  circuit  is  interlinked  with 
the  dielectric  and  magnetic  circuits.  Changes  in  the  cur- 
rent and  voltage  in  the  electric  circuit  are  accompanied  by 
changes  in  the  energy  stored  in  the  dielectric  and  magnetic 
fields,  thus  necessitating  a  readjustment  of  the  energy  con- 
tent in  the  whole  electric  system.  The  transfer  of  energy 
requires  time  and  thus  the  transient  period  is  of  definite, 
although  often  of  extremely  short,  duration. 

The  close  analogy  existing  between  electric,  dielectric 
and  magnetic  circuits  may  be  shown  to  advantage  by 
arranging  the  corresponding  quantities  in  tabular  form  as 
in  Table  II. 

For  convenience  in  solving  problems  the  energy  equations 
are  expressed  in  the  units  used  in  commercial  work: 
Energy  in  a  Magnetic  Field  : 

=  W  (joules)  =  ^en^)il(am_peres)     (lg) 

2i 
Energy  in  a  Dielectric  Field  : 

TTT/.     i    N        CYmicr  of  arads)  e2  (volts)      /irvv 

w  oouies)  =        -~ 


Energy  in  a  Moving  Body: 

=  TF(ergs)  =  M(grams)  ^  (meters  per  sec.) 

2i 
Energy  in  a  Moving  Body  : 

=  TF(joules)  =  M(kS')  v2  (meters  per  sec.) 

£ 
Energy  in  a  Moving  Body  : 


=  TFfft  Ib  }  =      -'    v"  (ft-_Per_sec.) 

2  X  32.2 
1   joule  =  1  watt-sec.  =  107  ergs  =  0.7376  ft.-lb. 

-  0.2389  g.-cal.  ='0.102kg.-m.  =  0.0009480  B.t.u.     (21) 
1  ft.-lb.  =  1.356  joules  =  0.3239  g.  =  0.1383  kg.-m. 

=  0.001285  B.t.u.  =  0.0003766  watt-hour     (22) 
1  B.t.u.  -  1,055  joules  =  778.1  ft.-lb.  =  252  g.-cal. 

-  0.2930  watt-hour     (23) 


INTRODUCTION 


TABLE  II 


Electric  circuit 

Dielectric  circuit 

Magnetic  circuit 

Electric  current: 

Dielectric     flux      (dielectric 

Magnetic  flux  (magnetic  cur- 

current): 

rent): 

i  =  Ge  =        electric   current. 

R 

*  =  Ce  =  *     lines    of    di- 

</>  =  Li   108   lines   of   mag- 

o 

netic  force. 

electric  force. 

Electromotive  force,  voltage: 

Electromotive  force: 

Magnetomotive  force: 

e  =  volts. 

e  =  volts. 

5   =  4-irni  ampere-turns. 

gilberts. 

Conductance: 

Condensance,      capacitance, 

Inductance: 

permittance  or  capacity 

n<j>      _        <b 

G  =       mhos. 
e 

C  =        farads. 

5            ™  » 

e 

henrys 

Resistance: 

Elastance: 

Reluctance: 

R  =  —  ohms. 

S  =  Q   =  ^  darafs. 

R   =    ,  oersteds. 

Electric  power: 

Dielectric  energy: 

Magnetic  energy: 

'P  =  R?'2  —  G€^  =   ic  watts. 

Ce2        tye 

T  i  2          (hi 

w  =       •    ==         joules. 

w  =  -        =   n   10~8  joules. 

£i                £ 

Electric-current  density: 

Dielectric-flux  density: 

Magnetic-flux  density: 

J  =       =  yG  amp.  per  cm.2 

D  =        =  KK  lines  per  cm.  - 
A. 

B   =    .  -  =  pH  lines  per  cm.2 

Electric  gradient: 

Dielectric  gradient 

Magnetic  gradient: 

G'  =      volts  per  cm. 

G'  =      volts  per  cm. 

/  =   -j  amp.  -turns  per  cm. 

Conductivity: 

Condensivity,      permittivity 

Permeability: 

j 

or  specific  capacity: 

7   =      mho.  -cm.3 

D 

B 

~  K 

M  =  H 

Resistivity: 

Elastivity: 

Reluctivity: 

1        G    , 
P  =        =       ohm-cm.3 

1        K 
k       D 

r   =  B 

Specific  electric  power: 
p  =  p/2  =  7G2  =  GI  watts 

Specific  dielectric  energy: 
kG'*       G'D. 

Specific  magnetic  energy- 

0.47TU/2           fB.» 

per  cm.3 


cm.3 

Condensance,     permittance, 
capacitance  current: 


d   =   ,,     =  C  -  -  amperes. 
at  at 

Dielectric-field  intensity: 
K  =  - — 2lines  of  dielectric 
force  per  cm.2 


w  =— ^  =  -10- 

joules  per  cm.3 
Inductance  voltage: 


Magnetic-field  intensity. 
//  =  47T/10-1  lines  of  mag- 
netic force  per  cm.2 


CHAPTER  II 
OSCILLOGRAPHS 

The  oscillograph  is  the  most  important  apparatus  for 
obtaining  quantitative  data  on  electric  transient  phenom- 
ena. To  gain  clear  concepts  of  the  relative  magnitude  of 
the  physical  quantities  involved  it  is  highly  desirable  for 


FIG.  7.— Magnetic  field  and  vibrating  elements. 

the  student  to  take  oscillograms  of  a  number  of  typical 
transients.  For  this  purpose  an  oscillograph  with  a  photo- 
graphic attachment  is  necessary. 

While  several  types  of  oscillographs  are  in  commercial 
use  all  operate  on  the  same  basic  principle.     The  essential 

10 


OSCILLOGRAPHS 


11 


element  of  the  oscillograph  is  the  galvanometer,  an  insu- 
lated loop  of  wire,  placed  in  a  magnetic  field,  through  which 
the  electric  current  flows.  The  direction  and  magnitude 
of  the  currents  cause  a  proportional  turning  movement  of 
a  small  mirror  attached  to  both  sides  of  the  loop.  The 
deflection  of  a  beam  of  light  thrown  on  the  mirror  indicates 
the  angular  position  of  the  mirror  and  hence  the  magnitude 
and  direction  of  the  current  flowing  through  the  loop. 

Three  Element  Oscillographs. — The  three  element,  port- 
able type  oscillograph  manufactured  by  the  General  Electric 
Co.  is  shown  in  Figs.  7  to  13.  The  arrangement  of  the 


Mirror- 


FIG.  8. — Vibrating  element. 


FIG.  9. — Cross  section  of  vibrating  element. 


electromagnetic  field  and  the  three  vibrating  elements  is 
shown  in  Fig.  7.  One  of  the  vibrating  elements  removed 
from  its  magnetic  field  is  shown  in  Fig.  8  and  its  vertical 
cross-section  in  Fig.  9.  The  three  vibrators  are  indepen- 


12 


ELECTRIC  TRANSIENTS 


dent  units  and  insulated  so  as  to  carry  three  separate  elec- 
tric currents.  The  vibrating  strips  and  mirrors  are  of 
silver.  The  vibrating  element  can  be  turned  around  a 
vertical  axis,  passing  through  the  center  of  the  mirror,  by 
the  screw  Q.  The  containing  cell  for  the  whole  vibrating 
element  is  also  movable  around  a  horizontal  axis,  passing 
through  the  center  of  the  mirror,  by  means  of  the  screw  S. 
Hence  the  beam  of  light  reflected  from  the  vibrating  mirror 
may  be  directed  to  any  desired  spot  and  so  adjusted  as  to 
pass  through  the  cylindrical  lens  to  the  slit  in  front  of  the 
rotating  photographic  film. 

In  Figs.  7  and  8,  the  letters  TT'  mark  the  terminals  of 
the  vibrating  strips  marked  ST  in  Fig.  9.  The  mirror 
with  the  vibrating  portion  of  the  loop  lies  between  the 
supports  BB1 '.  The  size  of  the  mirror  is  about  20  by  10  mils 
and  the  vibrating  element  has  a  natural  period  of  approxi- 


XA 

FIG.  10. — Optical  train— horizontal  projection. 


FIG.   11. — Optical  train — vertical  projection. 

mately  one  five-thousandth  of  a  second  (0.0002  sec.).  By 
immersing  the  vibrating  element  in  oil  the  instrument  is 
made  dead-beat. 


OSCILLOGRAPHS 


13 


The  horizontal  projection  of  the  optical  train  for  photo- 
graphic work  is  shown  in  Fig.  10  and  a  vertical  projection 
in  Fig.  11.  The  arc  lamp  is  at  A  and  the  arrows  indicate 
the  directions  of  the  beams  of  light.  PI,  P2,  P*  are  right- 
angled  prism  mirrors;  Si,  $2,  $3,  adjustable  slits;  h,  1%,  h 
condensing  lenses;  VMi,  VM^,  VMZ  the  vibrating  mirrors ; 
CL  a  cylindrical  lens  for  bringing  the  light  beams  to  a  sharp 
focus  on  the  photographic  film  on  the  surface  of  the  revolv- 
ing cylinder  in  the  film  holder. 


FIG.   12. — Oscillograph  on  operating  stand.      (Gen.  Elec.  Co.) 

In  Fig.  12  the  oscillograph  is  shown  mounted  on  a  con- 
veniently arranged  operating  stand.  The  positions  of  the 
arc  lamp,  film  motor,  film  holder,  controlling  rheostats, 
time  wave  oscillator  and  other  accessory  appliances  for 
recording  electric  transients  are  clearly  indicated.  The 
corresponding  wiring  diagram,  with  quantitative  circuit 
data,  is  shown  in  Fig.  13. 


14 


ELECTRIC  TRANSIENTS 


The  three-element,  portable  oscillograph  of  compact 
design,  manufactured  by  the  Westinghouse  Elec.  &  Mfg. 
Co.,  is  shown  in  Figs.  14  to  16.  The  photographic  film 
drum  and  driving  pulley  with  rheostats,  switches,  etc.,  are 
shown  on  the  right  side  of  Fig.  14 a,  while  on  the  left  are  the 


FIG.  13. — Wiring  diagram — three  element  oscillograph.     (Gen.  Elec.  Co.) 

three  sets  of  dial  resistances,  one  for  each  vibrating  element, 
with  switches,  binding  posts  and  fuses.  In  Fig.  146  is 
shown  the  driving  motor  with  control  apparatus  for  opera- 
ting the  film  holder  at  several  speeds.  Light  for  making 
the  photographic  record  is  obtained  from  a  low  voltage 
incandescent  lamp  of  special  design.  For  high  speed  records 
an  arc  lamp  is  used,  in  place  of  the  incandescent  lamp,  to 
gain  the  greatest  possible  light  intensity. 

The  galvanometer,  with  one  of  the  three  elements  re- 


OSCILLOGRAPHS 


15 


FIG.   14a. — Front  and  resistance  panel  side  of  portable  oscillograph.     (Westing- 
house  Elec.  &  Mfg.  Co.} 


FIG.  14&. — Front  view  of  portable  oscillograph  coupled  to  motor.     (Westinghouse 

Elec.  &  Mfg.  Co.) 


16  ELECTRIC  TRANSIENTS 

moved,  is  shown  in  Fig.  15.  The  moving  element  consists 
of  a  single  turn  or  oblong  loop  of  wire  forming  two  parallel 
conductors.  A  tiny  mirror  is  attached  to  both  conductors 
and  placed  in  a  strong  magnetic  field.  Hence  when  a 
current  passes  down  one  conductor  and  up  the  other,  one 
tends  to  move  forward  and  the  other  backward.  The 
mirror  bridging  these  conductors  is  given  an  angular  deflec- 
tion proportional  to  the  current. 


FIG.   15.- — -Three  element  galvanometer.     (Westinghouse  Elec.  &  Mfg.  Co.) 

The  design  of  the  electromagnetic  field  circuit  is  unique. 
A  direct  current  passing  through  a  single  coil  sets  up  a  mag- 
netic flux  which  passes  through  the  three  vibrating  elements 
in  series.  To  insulate  the  elements  from  each  other  and 
from  the  main  magnetic  core  and  yokes  four  insulating 
gaps  are  used,  thus  placing  seven  air  gaps  in  series  in  the 
path  of  the  magnetic  flux.  The  three  gaps  in  the  galvano- 
meter elements  are  J^2  in.  long,  giving  sufficient  space  for 
the  vibrators  and  producing  uniform  distribution  of  the 
magnetic  flux.  The  four  insulating  gaps  are  KG  in.  long 
but  of  large  cross-sectional  area  so  as  to  give  comparatively 
low  reluctance  in  the  magnetic  circuit.  The  field  excitation 
requires  6  volts,  direct  current. 

A  view  of  the  trip  magnet  and  shutter  release  mechanism 


OSCILLOGRAPHS 


17 


is  shown  in  Fig.  16,  in  which  the  trip  magnet  holds  the  long 
shutter  finger  so  that  the  short  finger  does  not  quite  touch 
the  shutter  tripping  arm.  The  shutter  is  a  tube  with  two 
opposite  longitudinal  slots.  The  tube  rotates  and  when 
the  slots  are  in  a  horizontal  plane  the  beams  of  light,  re- 
flected from  the  tiny  mirrors  of  the  galvanometer  vibrators, 
pass  through  the  cylindrical  condensing  lens  and  are 
focused  on  the  revolving  photographic  film.  This  occurs 
between  the  time  the  short  finger  falls  from  the  shutter 


FIG.  16. — Trip  magnet  and  shutter  release  mechanism.     (Westinghouse  Elec.  & 

Mfg.  Co.) 

tripping  arm  and  the  time  the  variable  finger  falls  from  the 
arm  one  revolution  later.  The  shutter  is  actuated  by  the 
spiral  spring  seen  just  beyond  the  finger  hub.  A  pin  on 
the  shutter  shaft  strikes  an  arm  on  the  lamp  extinguishing 
switch.  On  the  hub  are  attached  laminated  copper  strips 
which  complete  the  lamp  circuit  when  the  shutter  is  set  and 
which  break  the  circuit  when  the  shutter  snaps  closed. 
The  tripping  device  can  be  adjusted  so  as  to  start  exposures 
at  any  desired  part  of  the  film. 

Timing-waves  from  the  Oscillator  Alternator. — The 
time  factor  is  of  special  importance  in  electric  transient 
phenomena  and  some  means  for  recording  the  time  elapsed 


18 


ELECTRIC  TRANSIENTS 


is  necessary.  In  taking  oscillograms  in  which  the  transient 
current  or  voltage  recorded  does  not  give  directly  an  indica- 
tion of  the  time  consumed  it  is  customary  to  impress  an 
alternating  current  timing  wave  of  known  frequency  on 
one  of  the  vibrators. 

Current  for  the  timing  wave  may  be  takefh  directly 
from  any  available  power  circuit,  but  the  frequencies  of 
commercial  systems  are  to  some  extent  variable  and  the 
indicating  frequency  meters  may  not  be  sufficiently  accu- 


: — /VWK&&& — 


1ZOV.D.C 


FIG.  17. — Oscillator  alternator  circuit  diagram. 

rate  for  this  purpose.  A  convenient  source  of  supply  for 
timing  wave  current  of  constant  frequency  is  found  in  the 
oscillator  generator.  The  circuit  diagram  of  a  simple  porta- 
ble form  used  in  the  electrical  engineering  laboratories  of 
the  University  of  Washington  is  shown  in  Fig.  17.  The 
alternator  consists  of  an  audion  tube  connected  to  condens- 
ance,  resistance  and  inductance,  as  shown  in  the  circuit 
diagram,  of  the  following  quantitative  values: 

LI  =  0.756  henry s  R\  =  99  ohms 

L2  =  0.756  henrys  E2  =  99  ohms 

L(total)  ==  2.38  henrys  R3  =  96  ohms 

C  =  1.06  microfarads  R±  =  50  ohms 

The  impressed  d.c.  voltage  was  110  volts  but  other  values 
may  be  used  by  adjusting  the  resistance,  Rs.  The  ampli- 
tude of  the  timing  wave  may  be  varied  by  means  of  the 


OSCILLOGRAPHS 


19 


20  ELECTRIC  TRANSIENTS 

resistance,  R*.  The  alternating  current  produced  by  the 
oscillator  is  of  simple  sine  wave  form  and  has  a  constant 
frequency  of  100  cycles  per  second. 

Oscillograms. — Great  care  must  be  taken  in  making 
the  adjustments  on  the  oscillograph  in  order  to  produce 
good  oscillograms.  The  speed  of  the  revolving  drum  carry- 
ing the  sensitized  film  and  the  amplitude  of  the  galvano- 


FIG.   19. — Starting  transient  of  induction  motor  with  secondary  resistance  in 
circuit.     See  Fig.  167,  Chap.  IX. 

meter  mirror  vibrations  must  be  adjusted  to  meet  the 
conditions  imposed  by  the  transient  under  investigation. 
Thus  the  relative  drum  speed  for  the  oscillograms  shown 
in  Figs.  18,  19,  and  20  was  as  4  :  1  :  29,  and  the  amplitude 
adjusted  in  each  case  so  as  to  use  the  film  area  to  good 
advantage. 

The  time  lag  of  tripping  devices  and  shutter  operating 
mechanism  must  be  determined  so  as  to  expose  the  film  at 
the  instant  the  transient  occurs.  The  optical  train  must 
be  adjusted  so  as  to  give  a  spot  of  light  sharply  focused  on 
the  sensitized  film. 

Instructions  for  developing  the  films  and  for  printing  the 
oscillograms  are  given  in  the  Appendix.  A  circuit  diagram 


OSCILLOGRAPHS 


21 


22 


ELECTRIC  TRANSIENTS 


with  quantitative  data  should  be  attached  to  each  film. 
It  is  important  to  show  the  circuit  position  of  each  vibrator 


FIG.  21.- 


-Transfer  of  oscillating  energy  in  inductively   coupled   circuits. 
Chap.  VIII. 


See 


FIG.  22. — Same  circuit  as  in  Fig.  21.     Primary  opened  at  the  instant  all  the  oscil- 
lating energy  was  in  the  secondary  circuit. 

so  that  the  record  will  indicate  precisely  where  the  transient 
appearing  on  the  film  was  taken. 


OSCILLOGRAPHS 


23 


Problems  and  Experiments 

1.  Examine  the  oscillograph  with  care;  trace  all  the  circuits;  operate  the 
arc  lamp;  adjust  the  optical  train  until  the  mirror  on  each  vibrator  throws 
a  spot  of  light  through  the  slit  and  this  is  sharply  focused  on  the  ground 
glass  screen.  Arrange  a  circuit  with  variable  inductance  and  condensance 
as  indicated  in  Fig.  23.  Connect  vibrator  V\  by  means  of  a  shunt,  S,  so 


(VWWVW 


FIG.  23. 

as  to  indicate  the  current  wave  and  vibrator  Vz  with  a  resistance,  R\,  in 
series  to  show  the  voltage  wave.  By  means  of  the  small  synchronous  motor 
operate  the  large  oscillating  mirror  throwing  the  beams  of  light  on  the 
mica  screen  on  top  of  the  oscillograph.  By  varying  the  resistance,  induct- 
ance and  condensance  in  the  circuit  the  time  phase  relations  of  the  voltage 
and  current  may  be  changed  from  lag  to  lead. 

2.  Reproduce,  as  nearly  as  available  equipment  will  permit,  the  oscillo- 
gram  in  Fig.  18 


CHAPTER  III 
SINGLE  ENERGY  TRANSIENTS.     DIRECT  CURRENTS 

Transient  electric  phenomena  are  produced  by  changes 
in  the  magnitude,  distribution  and  form  of  the  energy 
stored  in  electric  systems.  The  simplest  types  of  electric 
transients  are  found  in  electric  circuits  having  only  one 
kind  of  energy  storage — that  is,  either  the  magnetic  or  the 
dielectric  field,  but  not  both.  A  condenser  discharging 
through  a  non-inductive  resistance,  as  illustrated  by  the 
oscillogram  in  Fig.  24,  gives  electric  transients  of  the 
simplest  type.  Since  the  resistance  in  the  circuit  is  con- 
stant the  current  is  at  all  instants  directly  proportional 
to  the  voltage  across  the  terminals  of  the  condenser.  The 
curve  on  the  oscillogram  can  therefore  be  used  as  represent- 
ing either  the  current-time  or  the  voltage-time  relation  as 
indicated  by  the  two  scales  in  the  figure. 

The  Exponential  Law. — The  energy  stored  in  the  con- 
denser is  at  any  instant  equal  to  Ce2/2.  The  rate  of 
discharge  is  ei,  which  must  be  equal  to  the  Ri 2  rate  of  energy 
dissipation  into  heat  in  the  resistance.  The  rate  of  energy 
discharge  is  therefore  at  any  instant  proportional  to  the 
energy  stored  in  the  condenser  and  the  rate  of  change  in  the 
current  is  at  any  instant  directly  proportional  to  the  magnitude 
of  the  current. 

Let  i  and  i'  represent  the  currents  at  any  two  points  on 
the  current-time  curve  of  the  oscillogram,  in  Fig.  24.  Then : 

di  di'  .  ., 

dt:dT::i:i  (24) 

Let  the  line  OP  be  drawn  from  starting  point  0  perpendi- 
cular to  the  X  axis.  Let  the  line  OQ  be  drawn  tangent  to 
the  curve  at  0  and  intersecting  the  X  axis  at  Q.  The  time 
represented  by  the  distance  PQ  is  called  the  time  constant, 

24 


DIRECT  CURRENTS 


25 


.s 


26  ELECTRIC  TRANSIENTS 

T,  of  the  circuit.     Since  i'  may  be  any  point  on  the  curve, 
let  it  be  taken  at  the  starting  point,  0 ;  then 

t'-70and      ^  =  -^  (25) 

From  (24)  and  (25) 

di  .      IQ  .  .  -  .  T 

df  ~T  •  •  %  ' Io  (26) 

Separating  the  variables  and  taking  the  limits  of  integra- 
tion from  the  starting  point,  0,  to  any  point  (i,  t)  oil  the 
curve : 


r*. 

JJ 


(27) 
(28) 

t  =  Joe-*  (29) 

Similarly,  for  the  corresponding  voltage-time  curve: 

-  F  f-'f  (30) 

e  —  J^Q€ 

Equations  (29)  and  (30)  show  that  the  fundamental 
relations  in  simple  electric  transients  are  expressed  by  the 
exponential  equation.  The  minus  sign  is  used  as  di/dt  is 
negative.  The  exponential  curve  represented  by  equations 
(29)  and  (30)  is  as  fundamental  in  the  study  of  electric 
transients  as  the  sine  wave  in  alternating  currents. 

If  the  energy  stored  in  a  magnetic  field  is  released  by 
short  circuiting  through  a  resistance  and  dissipated  into 
heat,  the  same  relations  exist.  Oscillograms  of  the  current- 
time  or  voltage-time  curves  similar  to  Fig.  24,  may  be 
obtained  by  discharging  a  magnetic  field  through  a  resis- 
tance, Fig.  27.  Likewise,  the  electric  transients  existing 
while  a  condenser  is  charged,  or  while  a  magnetic  field  is 
established,  obey  the  exponential  law.  In  Fig.  25  is  shown 
the  oscillogram  of  a  current-time  transient  obtained  while 
establishing  a  magnetic  field  in  the  circuit  shown  in  the 
diagram.  Let  the  line  OP  be  drawn  through  the  starting 
point  0  at  right  angles  to  the  X  axis.  Let  PS  be  drawn 


DIRECT  CURRENTS 


27 


parallel  to  the  X  axis  and  be  an  asymptote  to  the  current- 
time  curve.     Let  the  line  OQ  be  drawn  tangent  to  the  curve 


FIG.  25. — Forming  a  magnetic  field. 
E=  109  volts;/  -  0.36  amps.;  R  =  303ohms;L •  =  5.1  henrys;  T  =  0.017  seconds. 


/v 


/ 

!w 


FIG.  26. — Showing  permanent,  transient  and  instantaneous  values  for  oscillogram 

in  Fig.  25. 

at  0  and  intersecting  the  line  PS  at  Q.     The  line  OP  repre- 
sents the  value  of  the  permanent  current  /  which  is  equal 


28  ELECTRIC  TRANSIENTS 

to  E/R.  The  time  measured  by  the  line  PQ  is  the  time 
constant,  T,  of  the  circuit.  The  rate  of  storing  the  mag- 
netic energy  at  any  instant  is  proportional  to  the  remaining 
magnetic  storage  facilities  in  the  circuit  under  the  given 
conditions.  Therefore  the  rate  of  change  in  the  current  is  at 
any  point  on  the  curve  proportional  to  /  -  -  i,  and  equation 
(31)  is  derived  in  the  same  manner  as  equations  (24)  and  (26) 


(33) 

(34) 
The  transient  which  by  definition  represents  the  change 

from  one  permanent  condition  to  another  is  in  equation 

_  t 

(34)  represented  by  the  factor  —  Ie  T.  Before  the  circuit 
was  closed  the  value  of  the  current  was  zero,  while  the 
final  permanent  value  is  I. 

In  Fig.  26  the  permanent  or  final  value  of  the  current  is 
represented  by  OP,  the  distance  of  the  line  PS  from  the  X 
axis.  The  transient  values  are  given  by  the  ordinates  to 
the  broken  curve  WVX,  while  the  instantaneous  current 
which  must  at  any  instant  be  equal  to  the  algebraic  sum  of 
the  permanent  and  transient  values  is  given  by  the  ordi- 
nates to  the  curve  OYS,  which  is  the  curve  photographed 
on  the  oscillogram  in  Fig.  25.  It  is  important  to  note  that 
the  photographic  record  of  the  actual  instantaneous  values 
gives  at  each  point  the  resultant  or  the  algebraic  sum  of 
the  corresponding  permanent  and  transient  ordinates. 
Thus  for  any  time,  t: 

RY  =  RU  +  (-RV)  (35) 

The  Time  Constant.  —  From  the  starting  point  0  of  the 
current-time  curve  in  the  oscillogram,  Fig.  27,  which  shows 
the  discharge  of  a  magnetic  field  through  a  resistance,  the 
line  OP  is  drawn  at  right  angles  to  the  X  axis  ;  the  line  OQ  tan- 


DIRECT  CURRENTS 


29 


gent  to  the  curve  at  the  point  0  and  intersecting  the  X  axis  at 
Q'}  the  line  QN  perpendicular  and  ON  parallel  to  the  X  axis. 
From  the  principle  of  the  conservation  of  energy,  the 
energy  stored  in  the  magnetic  field  must  be  equal  to  the 
amount  dissipated  as  heat  in  the  resistance  of  the  circuit 
when  the  field  is  discharged. 


/»00  /» 

=  R  I    tfdt  =  RIQ2  I 

Jo  t/o 


RIJ-T 


(36) 


(37) 


In  circuits  having  resistance  and  inductance  in  series,  as 
in  Fig.  27,  the  time  constant  is  equal  to  the  inductance 
divided  by  the  resistance. 


FIG.  27. — Discharge  of  a  magnetic  field  through  a  constant  resistance. 
E  =  60  volts;  I  =  0.21  amps.;  R  =  28.6  ohms;  L=  0.89  henrys;  T  =  0.031 
seconds. 

In  a  similar  manner  the  expression  for  the  time  constant, 
T,  in  terms  of  the  circuit  constants  may  be  found  for  circuits 
having  condensance  and  conductance,  Fig.  24.  The  energy 
stored  in  the  condenser  when  the  discharge  starts  must  be 
equal  to  the  energy  expended  as  heat  in  the  Ri2  losses. 


30  ELECTRIC  TRANSIENTS 

CF  2  f ra  f ra       2t  J?T  2T 

^p_  =  R       tfdt  =  RI02       e~  Tdt  =  —^        (38) 

«/o  «/o 

Hence, 

T  =  CR  =  CG  (39) 

In  circuits  having  resistance  and  condensance  in  series, 
as  in  Fig.  24,  the  time  constant  is  equal  to  the  condensance 
divided  by  the  conductance. 

Equations  (37)  and  (39)  are  of  fundamental  importance 
in  the  study  of  transient  phenomena.  The  exponential 
equation  for  the  transients  in  Figs.  24  and  26  may  be 
rewritten  using  the  value  of  T  as  given  in  (37)  and  (39)  and 
the  data  in  the  circuit  diagrams. 

Fig.  24,  equations  (29),  (39) 

/  C1 

i  =  IQ€~T  =  7oc-c'  =  4.18€":  '"'amperes          (40) 
Fig.  24,  equations  (30),  (39) 

e  =  EQe~  *  =  E$.  =  120.6  e~38-5 Volts  (41) 

Fig.  25,  equations  (34),  (37) 

i  =  I  -  Ie~  T  =  I  -  7e~*'  =  0.36  -  0.36e~58'7  amperes  (42) 
The  reciprocals  of  the  time  constants  appearing  in  the 
exponential  equations  as  R/L  and  G/C  or  such  combinations 
of  circuit  constants  as  the  complexity  of  the  system  may 
require,  are  often  called  the  dissipation  constants  or  the 
attenuation  constants  of  the  circuit. 

The  expressions  for  the  time  constants  in  equations  (37) 
and  (39)  may  be  derived  from  the  current-time  and  voltage- 
time  curves  instead  of  basing  the  equations  directly  on  the 
principle  of  the  conservation  of  energy.  In  Fig.  24  the 
quantity  of  electricity  (coulombs)  in  the  condenser  when 
starting  the  transient  must  be  equal  to  the  total  amount 
expended  when  the  condenser  is  discharged,  as  represented 
by  the  area  between  the  current-time  curve  and  the  X  axis. 

rco  /»oo 

idt  =  70  I  e~  *  dt  =  IQT  (43) 


Jco  /» 

idt  =  70  1 
*/  o 


DIRECT  CURRENTS  31 

Hence, 

T  =  CET«  =  CR  =  Cn  =  0.026  seconds  (44) 

-/O  (JT 

Similarly,  in  Fig.  27,  the  magnetic  flux  in  the  field  when 
starting  the  transient  may  be  equated  to  the  total  number 
of  lines  of  force  cutting  the  circuit  when  all  the  magnetic 
energy  in  the  field  changes  into  heat  in  the  resistance. 


Hence, 


Jf-  r~        t 

edt  =  R\  idt  =  RIQ  I  e    T  dt  =  RI0T     (50) 
t/o  »/o 


T  =       =  0.31  seconds  (51) 

H 


If  the  initial  rate  of  discharge  in  Fig.  24  be  continued 
unchanged,  the  current-time  curve  would  coincide  with  the 
line  OQ  and  the  condenser  would  be  completely  discharged 
in  the  time  represented  by  PQ  or  T.  Hence  the  area  of  the 
rectangle  OPQN  must  be  equal  to  the  area  between  the 
current-time  curve  and  the  X  axis. 

Similarly,  in  Fig.  27,  if  the  initial  rate  of  discharge 
continued  unchanged,  the  current-time  curve  would  coin- 
cide with  the  line  OQ  and  all  the  energy  stored  in  the  mag- 
netic field  would  appear  as  heat  in  the  resistance  in  the  time 
represented  by  PQ  or  T.  Hence  the  area  of  the  rectangle 
OPQN  must  be  equal  to  the  area  between  the  current-time 
curve  and  the  X  axis. 

Expressions  for  the  transient  current  and  voltage  as  given 
in  (40),  (41)  and  (42)  are  derived  without  using  the  time 
constant  term.  The  customary  differential  equations 
giving  the.  basic  relations,  with  expressions  for  the  transient 
currents,  are  given  in  (52)  to  (55). 

For  circuits  having  resistance  and  condensance,  Fig.  24, 
while  the  condenser  is  discharged  through  a  constant 
resistance  : 

Ri  +    f-     ==  0;  hence  i  ==  /Oe~*c  ==  1$         (52) 


32  ELECTRIC  TRANSIENTS 

For  circuits  having  resistance  and  condensance,  similar 
to  Fig.  24,  the  transient  current  while  charging  from  0  to 
the  voltage  E  is  the  same  as  for  discharging  through  the 
resistance. 

Ri  +   (l~  ----  E',  hence  i  =  70e"«V  (53) 

For  circuits  having  resistance  and  inductance,  Fig.  19, 
while  the  magnetic  field  is  formed: 

Ri  +L-£  =  E-,  hence,  i  =  f  -  ^"^  =  /  -  /e~£  (54) 
at  K       H 

For  circuits  having  resistance  and  inductance,  Fig.  21, 
and  a  magnetic  field  supplied  by  a  current  70,  while  the 
field  discharges  through  a  short  circuit: 

Ri  +  L~  =  0;  hence,  i  =  70e~  ^  (55) 

Cvt' 

The  Exponential  Curve. — Oscillograms  of  simple  electric 
transients  give  a  photographic  record  of  the  current-time 
factors.  The  amplitude  of  the  curve  varies  directly  with 
the  magnitude  of  the  current  passing  through  the  vibrator 
and  the  strength  of  the  magnetic  field  in  which  the  vibrator 
moves.  The  length  of  film  used  for  any  given  unit  of 
time  depends  on  the  speed  of  the  revolving  drum  carrying 
the  film.  It  is  evident  that  both  the  amplitude  of  the 
mirror  vibrations  and  the  speed  of  the  film  may  be  adjusted 
independently  of  the  circuit  in  which  the  transient  occurs. 
By  examining  exponential  equations  representing  simple 
electric  transients  it  is  apparent  that  if  the  value  of  the  time 
constant,  T,  be  used  as  the  unit  of  length  on  the  X  axis  and 
the  initial  value  of  the  variable  as  the  unit  of  measure  for 
the  ordinates,  then  all  exponential  transients  will  have  the 
same  shape  and  may  be  represented  by  the  numerical 
values  of  the  exponential  equation,  y  =  e~x.  The  same 
space  unit  need  not  be  used  on  both  axes  to  represent  the 
unit  values  of  current  and  time,  but  the  scale  may  be 
selected  so  as  to  secure  a  convenient  shape  for  the  available 


DIRECT  CURRENTS 


33 


space.  In  Fig.  28  is  shown  a  current-time  curve  in  which 
the  unit  representing  the  initial  value  of  the  transient 
current  is  five  cm.,  while  the  unit  used  on  the  X  axis,  that 
is,  for  the  time  constant  of  the  circuit,  is  one  cm. 


FIG.  28. — The  exponential  curve.     Current-time  transient. 

By  using  the  initial  value  of  the  transient  as  the  unit  of 
ordinates  and  the  time  constant  of  the  circuit  as  the  unit 

TABLE  III 

y   =  e*;  €    =  2.71828 


y 


0.00                           1.000                          1.2 

0.301 

0.05 

0.951 

1.4 

0.247 

0.10 

0.905 

1.6 

0.202 

0.15 

0.860 

1.8 

0.165 

0.20 

0.819 

2.0 

0.135 

0.30 

0.741 

2.5 

0.082 

0.40                          0.670 

3.0 

0.050 

0.50 

0.607 

3.5 

0.030 

0.60                          0.549 

4.0 

0.018 

0.70 

0.497 

4.5 

0.011 

0.80 

0.449 

5.0 

0.007 

0.90 

0.407 

6.0 

0.002 

1.00 

0.368 

7.0 

0.001 

34  ELECTRIC  TRANSIENTS 

of  abscissae,  all  exponential  transients  are  of  the  same  shape 
and  if  plotted  to  the  same  scale  would  be  identical  with  the 
curve  in  Fig.  28.  The  numerical  relations  between  y  and 
x  in  the  exponential  equation  y  =  e~x  are  given  in  Table 
III.  While  the  plotting  of  transients  may  be  facilitated 
by  the  selection  of  the  above  units,  the  actual  initial  values 
of  the  transient  quantity,  expressed  in  amperes  or  volts, 
may  be  of  any  magnitude  as  determined  by  the  circuit 
conditions. 

Initial  Transient  Values. — In  simple  electric  transients 
the  initial  value  of  the  variable  quantity  depends  on  both 
the  permanent  value  and  on  the  relative  magnitude  of  the 
circuit  constants.  Thus  equations  (33)  and  (42)  show  that 
the  time  constant  of  a  magnetic  field  depends  on  the  induc- 
tance and  resistance  in  the  circuit.  If  the  energy  stored  in 
the  magnetic  field  be  discharged  by  short  circuiting  the 
terminals  of  the  field,  the  initial  value  of  the  transient  volt- 
age will  be  equal  in  magnitude  but  opposite  in  direction  to 
the  previously  permanent  value.  But  if  the  discharge  be 
made  through  an  additional  resistance,  R^  the  initial 
voltage  transient  will  be  greater  in  magnitude  in  the  ratio 
of  Ri  +  Rz  :  Ri,  when  Ri  represents  the  resistance  of  the 
field  winding.  The  time  constant  of  the  circuit  in  which 
the  transients  appear  would  be, 

T  =  D   when  the  field  is  short  circuited, 
HI 

and  TI  =  -D-~    '—=-  when  the  additional  resistance  #2  is 

Hi   T-   Hz 

inserted  in  the  discharging  circuit.  With  the  same  amount 
of  energy  stored  in  the  magnetic  field,  the  products  of  the 
initial  value  and  the  corresponding  time  constants  must 
be  equal. 

E0T  =  E0'T'  (56) 

Hence, 

EQ:EQf::Rl:R1  +  R2  (57) 

Eo>  =  EQR±±R*  (58) 

Hi 


DIRECT  CURRENTS 


35 


The  initial  induced  discharge  voltage  is  therefore  greater 
than  the  permanent  impressed  voltage  in  the  proportion 
of  the  resistances  in  circuit  for  the  two  cases.  In  the 
voltage-time  curve,  Fig.  29,  the  initial  discharge  voltage, 
Eo",  is  that  part  of  the  induced  voltage,  EQ',  due  to  the 
Ri2  drop. 


EV/ 
111       Q     = 


=  E 


R 


(59) 


This  relation  is  of  great  importance  in  the  design  and  opera- 
tion of  electrical  machinery.  In  breaking  electric  circuits, 
as  induction  coils,  motor  and  generator  fields,  transmission 


»  wwwwvwww 


FIG.  29. — Magnetic  field  discharging  through  additional  resistance,  Ri. 

lines,  etc.,  in  which  energy  is  stored  magnetically,  the  air 
or  oil  gap  in  the  switch  introduces  a  rapidly  increasing 
resistance.  The  faster  the  contact  points  of  the  switch  or 
circuit  breaker  separate,  the  more  rapidly  the  resistance  is 
inserted  and  the  higher  the  induced  voltage.' 

In  Fig.  30  is  shown  the  voltage-time  and  current-time 
oscillogfams  for  breaking  the  field  circuit  of  a  direct- 
current  motor.  In  opening  the  switch  an  arc  is  formed 
by  which  a  resistance  of  rapidly  increasing  magnitude  is 


36  ELECTRIC  TRANSIENTS 

introduced  into  the  circuit.  The  oscillograrn  shows  that  in 
about  >f  5  of  a  second  the  induced  voltage  increased  to  more 
than  twenty-eight  times  the  voltage  impressed  on  the  termi- 
nals of  the  field  before  the  switch  was  opened.  Although 
the  voltage  applied  to  the  motor  field  was  only  31.5  volts 
the  opening  of  the  switch  in  the  field  circuit  produced  a 
transient  stress  of  over  900  volts  on  the  field  insulation. 


FIG.  30. — Breaking  field  circuit  of  direct  current  motor.     Current  and  voltage 

transients. 

Since  the  transient  induced  voltage  on  the  motor  field 
winding  is  directly  proportional  to  the  rate  of  cutting  lines 
of  force  the  shorter  the  time  used  in  opening  the  switch, 
or  the  faster  the  resistance  is  inserted  in  the  circuit  the 
greater  the  transient  voltage-stress  tending  to  puncture  the 
field  insulation.  If  the  circuit  breaker  operates  in  steps  by 
which  resistances  of  known  value  are  introduced  into  the 
circuit  in  rapid  succession  the  transient  induced  voltage 
will  be  proportionately  lower  and  the  destructive  action 
of  the  arc  greatly  reduced.  Since  the  energy  stored  in  an 


DIRECT  CURRENTS  37 

electromagnetic  field  depends  on  the  current  flowing  in  the 
field  windings,  it  must  be  converted  into  some  other  form 
when  the  current  is  interrupted. 

Current,  Voltage  and  Magnetic  Flux  Transients. — In 
electromagnetic  circuits  having  constant  permeability  the 
current,  voltage  and  flux  transients  have  the  same  shape 
and  are  expressed  by  the  exponential  equation.  Referring 
to  Fig.  27 

e  =  Ri  (60) 

The  curve  in  the  oscillogram  therefore  represents  either 
the  current  or  voltage  transients  and  the  quantitative 
values  are  obtained  by  applying  the  corresponding  ampere 
and  volt  scales.  From  the  law  of  electromagnetic  induc- 
tion the  induced  voltage  is  equal  to  the  rate  of  cutting 
lines  of  force. 

e  =   .  10~8  volts  (61) 

at 

Hence, 

edt  = 


=  10s  I  e 


-Rt 


=  constant  e    L  lines  of  flux  (62) 

The  flux  transient  therefore  is  an  exponential  curve  of 
the  same  form  as  the  current  and  voltage  transients. 

In  Fig.  31  is  shown  the  corresponding  transients  for  the 
current,  voltage  and  flux  in  forming  an  electromagnetic 
field  in  a  magnetic  circuit  of  constant  permeability.  The 
transients  are  shown  by  the  dotted  lines,  the  permanent 
values  by  the  broken  lines  and  the  instantaneous  values 
by  the  full  line  curves. 


/  _  /€-  (63) 

/          _t\  _Rt 

=  E  +  \-Ee    T)  =  E  -  Ee   L  (64) 

_*4 
=  $  _  $e    L  (65) 


38 


ELECTRIC  TRANSIENTS 


T 


-/ 


/» 


L 


T 


1 


'N 


T 

$ 


\\  0 


1 


f N 


FIG.  31. — Single  energy  voltage,  current  and  magnetic  flux  transients  in  forming  a 
magnetic  field  in  a  magnetic  circuit  of  constant  permeability. 


DIRECT  CURRENTS  39 

Equations  (61),  (62)  and  (63)  express  the  instantaneous 
values  as  equal  to  the  algebraic  sum  of  the  permanent  and 
transient  quantities.  At  any  instant  in  time,  t,  as  indicated 
in  Fig.  31: 

PQ  =  the  instantaneous  value,  i,  e  or  <£  (66) 

PS  =  the  permanent  or  final  value,  /,  E  or  <£     (67) 

t  t  t 

PN  ••--  the  transient  value,  I0e~T,  EQe~T,  3>Qe~T    (68) 
In  discharging  energy  stored  in  a  magnetic  field,  in  a 
magnetic  circuit  of  constant  permeability,  through  a  con- 
stant resistance,  the  transient  and  the  instantaneous  values 
are  equal  as  the  permanent  value  is  zero. 

i  =  1^^  =  I^~Lt  (69) 

-'  -Rt 

e  =  E0e   T  =  E0e  L  (70) 

-'  -Rt 

0  =  $oe    T  =  $oe  L  (71) 

Problems  and  Experiments 

1.  A  condenser  of  115  mfds,  charged  to  500  volts,  is  discharged  through  a 
constant  resistance  of  425  ohms. 

(a)   Derive  the  equations  for  the  current-time  curve. 
(6)    Find  the  time  constant  of  the  circuit. 

(c)  Plot  the  voltage  across  the  terminals  of  the  condenser-time  curve. 
Ordinates  in  volts  and  abscissae  in  seconds. 

(d)  Draw  an  ampere  scale  of  ordinates  so  that  the  curve  plotted  in  (c) 
will  represent  the  current  transient. 

2.  The  time  constant  of  an  inductance  coil  is  found  by  taking  an  oscillo- 
gram  to  be  0.04  seconds.     The  resistance  in  circuit  was  15.8  ohms. 

(a)   Find  the  inductance  in  henrys. 

(6)  With  110  volts  impressed  on  the  coil  plot  the  starting  current 
transient. 

(c)    Write  the  equation  for  the  current-time  curve  in  (6). 

3.  Take  an  oscillogram  of  the  starting  current  transient  of  the  field  of  a 
laboratory  motor  or  generator. 

(a)  From  the  oscillogram  find  the  time  constant  of  the  field. 
(6)   Measure  the  resistance  of  the  circuit  and  calculate  the  inductance  of 
the  field. 

4.  With  the  vibrators  connected  as  shown  in  the  circuit  diagram  in  Fig. 
30  taken  an  oscillogram  showing  the  current  and  voltage  transients  pro- 
duced by  breaking  the  field  circuit  of  a  motor  or  generator. 

6.  By  means  of  oscillograms  determine  the  time  required  for  the  opera- 
tion of  automatic  circuit  breakers.  Arrange  the  connection  for  the  vibrators 
so  as  to  show  the  time  consumed  by  each  step  in  the  operation. 


CHAPTER  IV 

SINGLE  ENERGY  TRANSIENTS.     ALTERNATING 
CURRENTS 

In  direct-current  systems  the  transient  electric  phenom- 
ena described  in  the  preceding  chapter,  are  due  to  the 
storage  of  energy  in  magnetic  and  dielectric  fields.  If  a 
constant  direct-current  voltage  is  impressed  on  a  circuit 
having  constant  resistance  but  neither  inductance  or 
condensance  the  current  would  instantly  reach  its  perma- 
nent value,  and  any  change  in  the  voltage  would  at  the  same 
time  cause  a  proportional  change  in  the  current.  With 
either  condensance  or  inductance  in  the  circuit  a  short 
period  of  time  is  required  for  the  current  to  reach  its  perma- 
nent value  after  any  change  in  voltage,  and  the  current- 
time  curve  during  the  transition  period  is  expressed  by  the 
exponential  equation.  The  value  of  the  variable  current 
is  at  any  instant  equal  to  the  algebraic  sum  of  the  per- 
manent and  transient  values;  and  single  energy  transients 
for  direct  currents  in  circuits  having  constant  resistance 
and  inductance  or  constant  resistance  and  condensance, 
may  be  expressed  by  exponential  equations  similar  in  form 
to  (63),  (64)  and  (65). 

Single  Phase,  Single  Energy  Load  Circuit  Transients.— 
The  same  principle  applies  to  single  energy  transients  in 
alternating-current  systems.  In  circuits  having  constant 
resistance  but  neither  inductance  nor  condensance,  no 
transients  appear.  Any  change  in  the  voltage  produces 
instantly  a  proportionate  change  in  the  current.  In 
circuits  having  either  inductance  or  condensance  a  tran- 
sient period  for  the  readjustment  of  the  energy  content  of 
the  system  follows  any  change  in  voltage.  At  any  instant 
the  transient  current  or  voltage  is  the  algebraic  sum  of  the 

40 


ALTERNATING  CURRENTS 


41 


.  >  II 


42  ELECTRIC  TRANSIENTS 

corresponding  permanent  and  transient  values.  The  per- 
manent term,  i'  is  the  alternating  current  wave  assumed 
to  be  of  simple  sine  form  as  expressed  in  equation  (72)  with 
71  as  the  time  phase  angle  at  the  starting  moment. 

i'  =  +Vsin  (co£  -  7l)  (72) 

e'  =  +-#  sin  (co£  -  T2)  (73) 

The  transient  term,  i"  is  expressed  by  the  exponential 
equation  of  the  same  form  as  in  direct  currents  with  an 
initial  value  equal  in  magnitude  but  opposite  in  time  phase 
to  what  would  have  been  the  permanent  value  at  the  start- 
ing point  if  the  circuit  has  been  closed  at  some  previous  time. 

i"  =  ±  Vsin  7,  e~r  (74) 

e"  --    ±ME  sin  72  ^T  (75) 

The  instantaneous  value  of  the  current  or  voltage  would 
therefore  be  expressed  by  (76)  and  (77)  : 


i  =  t'  +  i"  a!  +  V  sin  (ut  -  71)  ±  "I  sin  71  e~*     (76) 
e  =  e'  +  e"  =  ±MI  sin  (ut  -  72)  +  "E  sin  Tl  e~T  (77) 

The  oscillogram  in  Fig.  32  shows  the  current-time  curve, 
i,  produced  in  a  circuit  having  20  ohms  resistance  and  0.575 
henrys  inductance  when  a  60  cycle  alternating-current  volt- 
age, e,  was  impressed  by  closing  a  switch  at  the  instant  in 
time  presented  by  the  OY  axis. 

The  voltage  impressed  is  represented  by  the  sine  wave, 

e  =  ME  sin  (co£  -  T2)  (78) 

The  actual  current  flowing  is  represented  by  the  curve 
i,  which  practically  coincides,  after  completing  6  cycles, 
with  the  permanent  value  i,  shown  by  the  dotted  sine  wave. 
The  transient,  i",  is  shown  as  the  broken  line  whose  initial 
value, 

ON  =  -OP  =  Vsin  Tl  (79) 

At  any  instant,  t,  after  the  closing  of  the  switch,  i  is  equal 
to  the  algebraic  sum  of  i'  and  i"  . 


ALTERNATING  CURRENTS 


43 


I  =  I   +  I 
=  0.835 


=  V  sin    co£  - 


wt  —  2 


+  0.835  sin 


-4 

'/  sin  Tl  e   L      (80) 

-35* 


(81) 


It  is  evident  that  the  initial  value  of  the  transient  may 
vary  from  -MI  to  +A7,  depending  at  what  point  of  the 
voltage-time  curve  the  circuit  is  closed.  If  the  switch  be 
thrown  at  the  instant  when  the  permanent  current  wave 
would  be  zero,  7  =  0,  no  transient  would  appear  and  the 


E 


FIG.  33. — Single  phase,  single  energy,  current  transient. 
129  volts;  ME  =  182.5  volts;  I  =  o'.59  amps.;  "I  —  .835  amps.; 
R  =  20  ohms;  L  =  0.575  henrys;  /  =  60  cycles;  71  =  0°. 


permanent  and  actual  current  time  curves  would  coincide 
throughout  as  shown  in  Fig.  33. 

i  =  i'  =  MI  sin  («0  (82) 

The  transient  current  would  have  a  maximum  initial 
value  if  the  circuit  is  closed  at  the  instant  the  permanent 
current  wave  is  at  a  maximum,  that  is  when  sine  71  =  90% 
The  time  constant  for  the  current  transient  would  be  the 


44  ELECTRIC  TRANSIENTS 

same  at  whatever  point  in  the  cycle  the  circuit  is  closed,  as 
it  depends  on  the  resistance  and  inductance  in  the  load 
circuit. 

Three-phase,  Single  Energy,  Load  Circuit  Transients.— 
For  three-phase  circuits  similar  relations  exist.     The  start- 
ing current   transients  in  three-phase   systems  in  which 
energy  may  be  stored  either  magnetically  or  dielectrically 
follow  the  same  laws  as  discussed  for  single-phase  circuits. 

In  Figs.  34  and  35  are  shown  oscillograms  of  the  three 
starting  load  currents  in  a  three-phase  system,  star  con- 
nected and  having  9.0  ohms  resistance  and  0.205  henrys 
inductance  in  each  phase  to  neutral.  The  corresponding 
permanent  current  waves  and  transient  currents  were 
traced  on  the  oscillogram  in  Fig.  34.  In  Fig.  35  the  circuits 
were  closed  at  the  instant  the  current  in  v\  was  of  zero 
value. 

Phase  1 : 

Impressed  voltage, 

61  =  "Ei  sin  (co£  —  72)  (83) 

Permanent  current, 

i'i  =  MIi  sin  (ut  —  71)  (84) 

Initial  value  starting  current  transient, 

OQ1  =  -OP,  =  "/sin  71  (85) 

Transient  current, 

_Ri 

i"i  =  "I i  sin  71  e    LI  (86) 

Actual  current,  oscillogram, 

^  =  i\  +  i'\  =  Vi  sin  (ut  -  71)  +  Vi  sin  yie  ^  (87) 
Time  constant, 

T  =  £-  =  0.023  seconds  (88) 

Phase  2: 

Impressed  voltage, 

e,  =  «E2  sin  (ut  -  72  -•  120°)  (89) 

Permanent  current, 

i\  =  «/2  sin  (ut  -  71  -  120°)  (90) 


ALTERNATING  CURRENTS 


45 


•IS 


46 


ELECTRIC  TRANSIENTS 


ALTERNATING  CURRENTS  47 

Initial  value  starting  current  transient, 

OQz  =  -OP2  =  "Iz  sin  (71  -  120°)  (91) 

Transient  current, 


t"2  =  V2  sin  (71  --  120°)«~5  (92) 

Actual  current,  oscillogram, 

i2  =  i'2  +  i"2  =  «/2  sin  (co£  -  72  --  120°) 


+V2  sin  (71  -•  12Q°)e~          (93) 
Time  constant, 

T2  =  f  2  =  0.023  seconds  (94) 

-^2 

Phase  3: 

Impressed  voltage, 

e,  =  MEz  sin  («*  --  72  -  240°)  (95) 

Permanent  current, 

i'8  =  "/8  sin  («*  --  71  -  240°)  (96) 

Initial  value  starting  current  transient, 

OQ3  =  -OP,  =  MIZ  sin  (71  -  240°)  (97) 

Transient  current, 

t"8  ==  "I*  sin  (71  --  240°)e  *•'  (98) 

Actual  current,  oscillogram, 

i3  =  i'8  +  i'%  =  V8  sin  (co^  -  71  -  240°) 

+  M/3  sin  (71  -  240°)  e~^        (99) 
Time  constant, 

Tz  =^  =  0.023  seconds  (100) 

/i3 

It  is  of  interest  to  note  that  the  sum  of  the  instantaneous 
values  of  the  three  currents,  ii  +  iz  +  i$,  is  equal  to  zero 
during  the  starting  period  as  well  as  after  the  permanent 
state  has  been  reached.  This  is  evidently  the  case  since 
under  permanent  conditions  the  sum  of  the  currents  is  at 
any  instant  equal  to  zero  and  hence  at  the  instant  the 
circuit  is  closed,  OPi  +  OP2  +  OP3  =  0.  Therefore,  the 
sum  of  the  initial  values  of  the  transient  currents,  OQi  + 
+  OQz  =  0,  and  as  the  time  constants  of  the  three 


48  ELECTRIC  TRANSIENTS 

transients  are  equal  the  sum  of  the  actual  currents  in  the 
three  phases  must  at  any  instant  be  equal  to  zero. 

Starting  Transient  of  a  Polyphase  Rotating  Magnetic 
Field. — In  the  preceding  illustrations  the  single  energy 
transients  are  due  to  changes  in  the  amounts  of  energy 
stored  in  the  given  circuits,  and  the  current-time  curves 
show  a  continuous  decrease  of  the  current  as  expressed  by 
the  exponential  equation.  If  the  permanent  condition 
relates  to  interconnected  circuits  which  permit  a  transfer 
of  energy  from  one  circuit  to  another,  although  the  total 
amount  of  energy  stored  in  the  magnetic  field  is  constant, 
as  in  the  rotating  field  of  a  polyphase  induction  motor, 
pulsations  will  appear  during  the  transition  period,  that 
merit  attention. 

In  Fig.  36  let  a  vector  of  constant  length,  ON,  rotating  in 
a  counter  clockwise  direction  represent  a  constant  rotating 
magnetic  field,  as  would  be  produced  by  three  equal  mag- 
netizing coils,  placed  120  deg.  apart,  and  excited  by  three- 
phase  currents,  as,  for  example,  in  a  three-phase  induction 
motor.  For  simplicity  let  the  rotor  be  removed  and  con- 
sider the  stator  circuit  and  the  magnetic  flux  during  the 
starting  transition  period  in  which  the  rotating  field  is 
built  up  to  its  constant  permanent  value.  Let  the  switch, 
connecting  the  stator  circuit  to  three-phase  mains,  be 
closed  at  the  instant  the  rotating  magnetic  flux  vector,  ON, 
lies  along  the  X  axis,  ON0  in  Fig.  36 ;  as  would  have  been  the 
case  if  the  switch  had  been  closed  at  some  previous 
time.  The  actual  value  of  the  rotating  flux  at  the  instant 
the  circuit  is  closed  is  zero.  The  permanent  value  is 
represented  by  ONQ  and  since  the  initial  value  of  the  transi- 
ent flux  must  be  equal  in  magnitude  but  of  opposite  time 
phase,  it  is  represented  by  the  vector  OQ0. 

OQQ  =--  (-(W0)  (101) 

From  its  initial  values  OQ0  the  transient  flux  decreases 
in  magnitude,  as  indicated  by  the  exponential  flux-time 
curve  in  Fig.  37,  but  continues  fixed  in  space  direction 


ALTERNATING  CURRENTS  49 

along  the  X  axis.  After  the  time,  ti,  has  elapsed,  repre- 
sented by  the  time  angle  N0ONi,  the  transient  has  a  value 
OQi.  The  actual  value  of  the  flux  must  be  the  vector  sum 
of  the  permanent  value  ONi  and  the  transient  OQi,  or  the 
resultant  OPi.  In  the  time  t2,  the  transient  has  decreased 
to  OQ2  and  the  permanent  flux  vector  reached  the  position 
ON2.  The  actual  flux  OP2  is  the  resultant  of  OQ2  and  ON2. 
Similarly  OP3  is  the  resultant  of  OQ3  and  ONZ]  OP4,  of 


FIG.  36. — Permanent,  transient  and  instantaneous  values  of  the  magnetic  flux  in 
starting  a  rotating  magnetic  field.     Polar  coordinates. 

OQ4  and  ON^  etc.  From  the  vector  diagrams,  Figs.  36 
and  38,  it  is  evident  that  the  actual  starting  flux  will  oscil- 
late having  values  greater  and  smaller  than  the  permanent 
value,  the  number  of  oscillations  depending  on  the  time 
constant  of  the  circuit.  The  maximum  value  of  the  flux 
in  the  starting  period  will  in  any  case  be  less  than  double  the 
permanent  value,  as  the  transient  flux  continuously 
decreases  from  an  initial  value  equal  to  the  permanent  flux 
in  magnitude  and  different  by  180  deg.  in  time-phase. 


50 


ELECTRIC  TRANSIENTS 


The  flux-time  curve  in  Fig.  39  gives  in  rectangular  coordi- 
nates the  same  relation  as  shown  by  the  flux  vector  OP  in 
the  polar  diagram  in  Fig.  38. 

Polyphase  Short  Circuits.  Alternator  Armature  and 
Field  Transients. — Consider  a  three-phase  alternator  carry- 
ing a  constant  balanced  load  of  constant  power  factor. 
The  three-phase  currents  flowing  in  the  armature  produce 
a  resultant  constant  armature  flux  or  armature  reaction. 


FIG.  37. — Starting  magnetic  flux  transient  from  Fig.  36.     Rectangular  coordi- 
nates. 

With  respect  to  the  field  the  armature  flux  is  stationary  but 
with  respect  to  any  diameter  of  the  armature  taken  as  a 
reference  axis,  the  armature  currents  produce  a  constant 
rotating  field  of  the  same  nature  as  the  constant  rotating 
field  of  a  three-phase  induction  motor.  For  a  machine  in 
which  the  field  is  on  the  rotating  spider  while  the  armature 
is  stationary,  the  resultant  flux  producing  armature 
reaction  rotates  synchronously  with  the  field.  For  an 
alternator  with  the  field  stationary  and  the  armature 
rotating  the  resultant  constant  flux  rotates  at  the  same 


ALTERNATING  CURRENTS 


51 


speed  but  in  direction  opposite  to  the  rotation  of  the  arma- 
ture and  therefore  is  stationary  with  respect  to  the  frame 
of  the  machine.  In  either  case  the  armature  flux  or  the 
armature  reaction  is  stationary,  if  referred  to  the  alternator 
field,  but  is  a  synchronously  rotating  field  with  respect  to 
the  armature. 


FIG.  38. — Starting  a  rotating  magnetic  field.     Polar  vector  diagram  of  magnetic 
flux  for  three  cycles  of  Fig.  36. 

Due  to  the  close  proximity  of  the  armature  conductors 
to  the  field  poles  a  large  part  of  the  magnetic  flux  produced 
by  the  armature  currents  passes  through  the  field  magnetic 
circuit.  This  causes  a  reduction  in  the  field  flux  and 
therefore  in  the  amount  of  energy  stored  magnetically  by 
the  exciting  current  in  the  field.  Hence,  although  a  con- 
stant direct-current  voltage  is  impressed  on  the  field  circuit 
the  useful  flux  is  greatly  reduced  by  the  armature  reaction, 
and  as  a  consequence  the  generated  armature  voltage 
decreases  in  the  same  proportion.  To  effect  any  change  in 
the  amount  of  energy  stored  magnetically  takes  time  and 
therefore  the  interaction  of  the  armature  flux  with  the  field 
magnetic  circuit  produces  electric  transients. 

During  the  transition  period  following  the  instant  the 
short  circuit  occurs,  two  distinct  causes  are  therefore 
superimposed  in  producing  transient  phenomena  in  the 


52 


ELECTRIC  TRANSIENTS 


interlinked   electric   and   magnetic   circuits   of   polyphase 
alternators. 

(a)  The  armature  transient  which  is  equivalent  to  the 
starting  transient  of  a  rotating  magnetic  field  including  full 
frequency  pulsations  as  illustrated  in  Figs.  38,  39. 

(b)  A  field  transient  due  to  the  reduction  of  the  field  flux 
by  the  armature  reaction. 


FIG.  39. — Same  data  as  in  Fig.  38.     Rectangular  coordinates. 

The  transients  produced  under  (a)  and  (b)  differ  in 
duration,  the  ratio  being  in  each  case  determined  by  the 
relative  time  constants  of  the  armature  and  field  circuits. 
In  general  the  time  constant  in  the  field  circuit  is  greater 
than  in  the  armature  circuits.  Large  turbo-alternators 
have  very  slow  field  transients  as  compared  to  the  duration' 
of  the  armature  transients. 

In  Fig.  40  is  shown  an  oscillogram  of  transients  produced 
by  a  short  circuit  on  all  three  phases  of  a  7.5  kw.,  240  volt, 
60  cycle,  three-phase,  star-connected  alternator  running 
idle  and  with  40  per  cent  normal  field  excitation.  A  similar 
oscillogram  of  short  circuit  transients  for  the  same  machine 
while  carrying  50  per  cent  of  full  load  is  shown  in  Fig.  41. 


ALTERNATING  CURRENTS  53 

As  indicated  in  the  circuit  diagrams,  Figs.  40  and  41, 
vibrator  Vi  records  the  armature  voltage  across  one  pair  of 
slip  rings,  ea\  vibrator  i>2,  the  current,  ia,  in  one  armature 
circuit,  and  vibrator  vs  the  field  current,  if.  As  the  short 
circuit  is  directly  across  v\,  the  voltage  ea  instantly  drops  to 
zero.  The  transient  in  the  field  winding  is  due  to  the  com- 
bined action  of  the  starting  transient  of  the  rotating  field 


FIG.  40. — Short  circuit  transients  from  no  load.     Three-phase  alternator,  star- 
connected. 

E,  no  load  =  109  volts;  7,  short  circuit  =  12.0  amps.;  I,  field  =  1.25  amps.; 

/  =  60  cycles. 

in  the  armature,  which  produces  the  full  frequency  pulsa- 
tions, and  the  slower  field  transient  resulting  from  the 
reduction  of  the  field  flux  by  the  armature  reaction.  In 
breaking  the  short  circuit  the  field  transient  alone  will 
appear  in  the  field  winding,  as  shown  by  the  oscillograms 
in  Figs.  42  and  43.  Necessarily  the  transient  is  reversed 
in  direction  from  what  is  represented  in  Figs.  40  and  41, 
when  the  short  circuit  is  made.  It  should  be  noted  that 
breaking  the  armature  short  circuit  was  not  instantaneous 


54 


ELECTRIC  TRANSIENTS 


a 

o  ^ 


II 

§ 


ALTERNATING  CURRENTS 


55 


O    ^H 
.^   CO 


bC  3 

a 


56 


ELECTRIC  TRANSIENTS 


ALTERNATING  CURRENTS 


57 


as  arcs  formed  at  the  switch  and  continued  the  circuit 
during  the  time,  a  —  b,  Fig.  42,  approximately  for  %  of 
a  cycle  or  J^oo  °f  a  second.  During  this  period  the 
energy  stored  magnetically  in  the  armature  circuits  was 
dissipated.  Much  more  time,  over  10  complete  cycles, 
was  required  to  restore  full  excitation  in  the  field  poles. 


FIG.  44. — Armature  current  transients.     Short-circuit  on  three-phase,  star-con- 
nected alternator,  no  load. 

E  =  280  volts;  /,   short  circuit  =  28.7  amps.;  7,  field  =  3.3  amps.;  /  =  59 
cycles. 

The  direct  relation  of  the  voltage  generated  in  the  arma- 
ture to  the  variable  useful  field  flux  is  shown  by  the  voltage 
wave,  ea,  and  the  field  transient,  if,  in  Fig.  42.  When  the 
short  circuit  is  made  the  same  transients  occur,  but  reversed 
in  time,  as  is  evident  from  oscillograms  in  Figs.  40,  41,  44 
and  45. 

In  the  operation  of  alternators  the  relative  value  of  the 
initial  or  momentary  to  the  final  or  permanent  short  circuit 
currents  is  of  great  importance.  At  any  instant  the  short 
circuit  current  obeys  Ohm's  law,  that  is  in  magnitude  it 


58  ELECTRIC  TRANSIENTS 

will  be  directly  as  the  voltage  generated  and  inversely  as 
the  impedance  of  the  armature  circuit. 

ia   =    ^   ••--~^-^-'~x-  (102) 

Since  the  armature  resistance,  Ra,  is  small  compared  to 
the  armature  reactance  Lxa,  equation  (102)  may  be  written 
as  in  (103). 

ia  =•-  ~  (103) 

For  constant  speed  the  generated  voltage,  e0,  is  directly 
proportional  to  the  useful  flux.  At  the  instant  the  short 
circuit  occurs  and  the  alternator  carries  no  load,  as  in  Fig. 
40,  the  useful  flux  depends  on  the  direct  current  voltage 
impressed  on  the  field  winding  and  produces  an  armature 
voltage,  Oea,  and  hence  the  initial  or  momentary  value  of 
the  short  circuit  current, 

Oia  =  -?-a-  (104) 

joXa 

If  expressed  in  effective  values  as  if  the  current  sine  wave 
continued  at  the  initial  magnitude, 

./.  =  --  (105) 

ifi  a 

During  the  transient  period  following  the  short  circuit 
the  armature  reaction  reduces  the  field  flux  and  as  a  conse- 
quence the  voltage  generated  in  the  armature  decreases 
in  the  same  ratio.  With  the  expiration  of  the  field  tran- 
sient the  useful  flux,  <j>u,  is  constant  and  hence  the  gener- 
ated voltage,  Ea,  and  the  armature  current,  Ia,  are  constant 
or  have  permanent  values. 

Ia    ==    —  (106) 

I a        Ea        $M  _    field  excitation — armature  reaction 
0Ia       0Ea       0&u  field  excitation 

(107) 

Although  the  decrease  in  the  armature  current  from  its 
initial  to  the  permanent  value,  as  shown  in  Figs.  44  and  45, 


A  L  TERN  A  TING  C  URREN  TS 


59 


is  due  to  a  reduction  in  the  useful  field  flux  and  hence  in  the 
generated  armature  voltage  it  is  customary  to  consider  the 
voltage  constant  and  ascribe  the  change  to  a  fictitious 
increase  in  the  reactance  of  the  armature  circuit.  The 
combined  effect  of  the  armature  reaction  and  the  true  arma- 
ture reactance  is  represented  by  the  so-called  synchronous 
reactance  sxa. 


FIG.  45. — Armature    current    transients.     Short-circuit    on    three-phase,     star- 
connected  alternator,  50  per  cent,  of  full  load. 

The  permanent  short  circuit  current  may  therefore  be 
expressed  by  equation  (108)  and  the  ratio  of  the  permanent 
to  the  initial  or  momentary  values  by  (109) 

Let :  Ia  =  permanent  short  circuit  armature  current. 
0I a  =  initial  short  circuit  armature  current. 
Lxa  =  armature  reactance. 

sxa  =  synchronous  reactance  =  armature  reactance 
+  armature  reaction. 


T  °a 

a  ~      r 

S'^ffl 

J-  a     L%a 

a*  a  sX  a 


(108) 
(109) 


60  ELECTRIC  TRANSIENTS 

If  it  be  assumed  that  the  permeability  of  the  magnetic 
circuits  remains  constant  for  the  changes  in  flux  density, 
the  field  current-time  curve  may  also  be  expressed  in  the 
form  of  an  equation  in  terms  of  the  circuit  constants  and 
the  initial  value  of  the  transients. 

Let,      Rf=  resistance  of  field  circuit. 
Lf=  inductance  of  field  circuit. 
Ra=  resistance  of  armature  circuit. 
La=  inductance  of  armature  circuit. 
t=  time  from  the  instant  short  circuit  occurs. 
oo  =  27r/;  /  =  frequency  in  cycles  per  second. 
i/=  instantaneous  value  of  field  current. 
//=  permanent    value    of    field    exciting    current 

before  short  circuit  occurs. 
i'/=  instantaneous  value  of  current  in  field  circuit 

due  to  field  transients. 
/'/  —  initial  value  of  i'  f. 
i'af=  instantaneous  value  of  current  in  field  circuits 

due  to  armature  transient. 
I'  af=  initial  value  of  i'  '  af 


(110) 


-R*t 


i'af  =  raf€         sin  («0  (111) 

In  Figs.  42,  43: 


if=If  -  i' s  =  If  -  rfe   J  (112) 

In  Figs.  40,  41: 

iaf=  if+i'f  +  i'af  -.=  J  +  rr^V/'e'^'sinarf  (113) 

Short  circuit  currents,  particularly  under  normal  field 
excitation,  produce  so  great  changes  in  flux  density  that 
the  permeability  is  not  constant  and  hence  La  and  Lf  are 
not  constant.  The  purpose  of  the  equation  is  however, 
merely  to  state  in  concise  form  the  factors  involved  without 
taking  into  consideration  the  complications  due  to 
variations  in  the  permeability  of  the  magnetic  circuits. 


ALTERNATING  CURRENTS 


61 


While  short  circuits  produce  electric  transients  of  greater 
magnitude  than  the  changes  that  occur  during  normal 
operation  of  alternators,  it  should  be  kept  in  mind  that  any 
modification  in  the  armature  currents,  as,  for  example,  an 
increase  or  decrease  in  the  load,  produces  transients  having 
the  same  characteristics  as  those  produced  by  short  circuits. 
Any  change  in  the  amount  of  energy  stored  magnetically 
in  the  armature  or  field  circuits  requires  time  and  during 
the  period  of  readjustment  electric  transients  are  produced 
in  the  interlinked  electric  and  magnetic  circuits. 


FIG.  46. — Short  circuit  transients,  single  phase  alternator.     Symmetrical.     No 

load. 

E  =  108    volts;  7,   short   circuit  =  10.4  amps.;  /,  field  =  2.6   amps.;  /  =  60 
cycles. 

Single -phase  Short  Circuits.  Alternator  Armature  and 
Field  Transients. — In  polyphase  alternators  the  permanent 
armature  field  produced  by  the  balanced  armature  currents, 
and  hence  the  armature  reaction,  is  constant  in  value  and, 
with  respect  to  the  alternator  field  poles,  fixed  in  position. 
In  single-phase  alternators  the  magnetic  field  produced  by 
the  armature  currents,  and  therefore  the  armature  reaction, 


62  ELECTRIC  TRANSIENTS 

pulsates  synchronously  with  the  armature  rotation.  The 
pulsations  of  the  armature  reaction  necessarily  appear  in 
the  field  circuit.  As  the  armature  rotates  180  electrical 
degrees  for  each  half  cycle  of  the  armature  current,  the 
pulsations  of  the  armature  reaction  with  respect  to  the 
field  poles  will  have  double  the  frequency  of  the  armature 
currents.  Therefore,  the  field  current  has  a  permanent 
double  frequency  pulsation  as  shown  in  Figs.  46  and  47. 


FIG.  47. — Short  circuit  transients,  single  phase  alternator.     Symmetrical.     Load. 
E,  load  =  106  volts;  /,  load  =  16.83  amps.;  /,  short  circuit  =  21.5  amps.; 
7,  field  =  2.6  amps.;  /  =  60  cycles. 

Since  the  armature  reaction  is  pulsating  and  not  constant, 
as  in  polyphase  alternators,  the  initial  value  of  the  starting 
transient  of  the  armature  flux  will  depend  on  the  point  on 
the  current  wave  at  which  the  short  circuit  occurs.  Thus 
in  Figs.  46  and  47  the  short  circuiting  switch  closed  nearly 
at  the  instant  the  armature  current  was  zero  and  hence 
only  a  very  small  armature  transient  was  produced.  With 
the  armature  transient  absent  the  field  current-time  oscit- 
lograms,  as  illustrated  in  Figs.  46  and  47,  are  symmetrical 


ALTERNATING  CURRENTS  63 

showing  the  permanent  double  frequency  pulsations  of  the 
armature  reaction  superimposed  on  the  field  transient. 
If  the  short  circuit  occurs  at  other  than  the  zero  points 
on  the  armature  current  wave,  an  armature  transient  of  full 
frequency  is  produced  for  the  same  reason  as  explained  for 
short  circuits  in  polyphase  alternators.  The  oscillograms 
in  Figs.  48  and  49  show  the  asymmetrical  field  current- 


FIG.  48. — Short  circuit_transients,  single  phase  alternator.     Asymmetrical.     No 

load. 

E  =  57  volts;  /,   short   circuit  =  23.0   amps.;   I,   field  =  1.3   amps.;  /  =  60 
cycles. 

time  curves  on  which  are  superimposed  the  double  fre- 
quency permanent  armature  reaction,  the  field  transient, 
and  the  full  frequency  pulsation  produced  by  the  armature 
transient.  The  combination  of  the  full  frequency  arma- 
ture transient  pulsation  with  the  permanent  double 
frequency  armature  reaction  produces  the  asymmetry 
in  the  curves.  The  ordinates  for  the  odd  numbers  of  the 
double  frequency  waves  add  to  the  full  frequency  values, 
while  for  the  even  number  of  waves  the  difference  in  the 
ordinates  produces  the  wave  recorded  by  the  oscillograph. 


64 


ELECTRIC  TRANSIENTS 


ALTERNATING  CURRENTS  65 

Hence,  during  the  transition  period  the  peaks  of  the  odd 
numbered  waves  decrease,  while  the  even  numbered  peaks 
increase,  and  at  the  expiration  of  the  armature  transient 
all  reach  the  permanent  constant  pulsation  produced  by 
the  pulsating  armature  reaction.  While  the  field  current 
pulsates  as  a  result  of  the  double  frequencyarmature  reac- 
tion and  the  full  frequency  armature  transients,  the  voltage 
across  the  field  terminals  will  pulsate  to  a  greater  or  less 
degree  depending  on  the  amount  of  external  resistance  and 
inductance  in  series  with  the  field  circuit.  With  much  ex- 
ternal resistance  or  impedance  the  voltage  at  the  terminals 
of  the  field  winding  may  reach  high  values  which  may 
puncture  the  insulation  and  cause  a  short  circuit  in  the 
field  exciting  circuit. 

The  field  transient  separated  from  the  armature  reaction 
may  be  shown  by  taking  an  oscillogram  of  the  field  current 
when  the  short  circuit  on  the  single  phase  alternator  is 
broken,  as  shown  in  Figs.  50  and  51.  The  armature  tran- 
sient is  dissipated  during  the  opening  of  the  switch,  indi- 
cated by  the  time  a  —  b  on  the  oscillogram,  while  several 
complete  cycles  are  required  before  the  field  flux,  and  as  a 
consequence  the  armature  voltage,  regains  its  full  value. 
In  the  transition  period  following  the  closing  or  opening  of 
the  short  circuiting  switch  the  oscillograms  of  the  field 
currents  show  the  effects  of  the  energy  changes  taking 
place  in  both  the  field  and  armature  circuits. 

Under  the  assumption  that  the  permeability  of  the  mag- 
netic circuits  is  constant  the  field-current-time  curve  in 
Figs.  46  to  51  may  be  expressed  in  terms  of  the  circuit 
constants  and  the  initial  values  of  the  transients : 
Let:    Rf  =  resistance  of  field  circuit. 

Lf  =  inductance  of  field  circuit. 

Ra  =  resistance  of  armature  circuit. 

La  =  inductance  of  armature  circuit. 
t  =  time  from  the  instant  short  circuit  occurs. 
co  =  27r/;  /  =  frequency  in  cycles  per  second. 
if  =  instantaneous  value  of  field  current. 


66 


ELECTRIC  TRANSIENTS 


ALTERNATING  CURRENTS 


67 


J3  05 

a-1 

83 


68  ELECTRIC  TRANSIENTS 

I  f  =  permanent   value    of   exciting   current   before 

short  circuit  occurs. 
i'  f  =  instantaneous  value  of  current  in  field  circuit 

due  to  field  transient. 
/'/  =  initial  value  of  if  /. 
iaf  =  instantaneous  value  of  current  in  field  circuit 

due  to  armature  reaction. 
la/  =  maximum  value  of  ia/. 
if  af  =  instantaneous  value  of  current  in  field  circuit 

due  to  armature  transient. 
Iaf  =  maximum  initial  value  of  i'  af. 

71  =  phase  angle  of  iaf. 

72  =  phase  angle  of  i'  «/. 


iaf  =  MIa/'  sin  (2 cot  —  71)  (115) 

i 
In  Fig.  50: 


_RO( 
'af  =  "I'af6  La  sin  (ut  -  72)  (116) 


-£i 


i,  =  //  -  7/e    L'  (117) 

In  Fig.  46: 

if  ----  If  +  rf*  L'  +  "Ja/  sin  (2^  -  71)         (H8) 
In  Fig.  48: 


-Rft 


if  =  If  +  rfe   L!  +  MIaf  sin  (2wf  --  71) 

t 

««  -7a)  (119) 


-R"t 


As  indicated  by  the  difference  in  the  upper  and  lower 
halves  of  the  double  frequency  pulsation  the  permeability 
of  the  magnetic  circuit  changed  with  the  flux  density. 
Under  full  field  excitation  the  short  circuit  transients 
would  produce  much  greater  changes  in  the  flux  density 
and  hence  in  the  permeability  of  the  steel  in  the  armature 
and  field  poles.  For  this  reason  the  equations  are  not 
directly  applicable  to  commercial  problems  but  state  the 


ALTERNATING  CURRENTS 


69 


relations  of  the  factors  involved  provided  the  permeability 
of  the  iron  core  is  constant. 

Single -phase  Short  Circuit  on  Polyphase  Alternators.— 
If  all  phases  of  polyphase  alternators  are  short  circuited 
simultaneously  the  armature  transients  appear  in  the  field 
circuit  as  full  frequency  pulsations  produced  by  the  rotating 
magnetic  field,  as  illustrated  for  three-phase  machines  in 
Figs.  40  to  43. 


FIG.  52. — Single  phase  short  circuit  on  three  phase  alternator. 

If  one  phase  only  is  short  circuited  the  effect  on  the  field 
circuit  is  essentially  the  same  as  illustrated  for  single  phase 
alternators  in  Figs.  46  to  48.  In  Fig.  52  is  shown  the 
transient  of  the  field  current  of  a  three-phase  alternator 
after  short  circuiting  one  phase.  The  field  current-time 
curve  shows  the  effects  produced  by  the  field  and  armature 
transients  and  the  permanent  double  frequency  pulsations 
due  to  the  armature  reaction.  In  Fig.  53  is  shown  an 
oscillogram  for  a  single-phase  short  circuit  on  a  three-phase 
alternator  which  after  4  cycles  is  followed  by  a  short  circuit 


70 


ELECTRIC  TRANSIENTS 


on  all  three  phases.  While  only  one  phase  is  short  circuited 
the  field  current  shows  the  double  frequency  pulsations 
combined  with  both  the  armature  and  field  transients. 
After  the  three-phase  short  circuit  occurs  the  field  current 
shows  the  full  frequency  pulsations  of  the  armature  tran- 
sient combined  with  the  slower  field  transient. 


FIG.  53. — Single  phase  short  circuit  on  three  phase  alternator  followed  by  a  three 

phase  short  circuit. 

E  =  118  volts;  7,   load   =10  amps.;  /,  short  circuit  =  22.5  amps.;  I,  field  = 
2.6  amps. 

Oscillograms  of  transients  in  polyphase  systems  produced 
by  single-phase  short  circuits  necessarily  differ  with  the 
type  of  machine  and  the  way  the  transient  magnetic  fluxes 
interlink  with  the  electric  circuit  to  which  the  oscillograph 
vibrator  is  connected.  Thus  in  Fig.  54  the  open  phase 
voltage  of  a  two-phase  alternator  with  short  circuit  on  one 
phase  shows  a  triple  frequency  harmonic,  while  the  field 
current  shows  the  double  frequency  pulsation  combined 
with  the  field  and  armature  transients  of  the  same  charac- 
teristics as  for  single-phase  alternators. 


ALTERNATING  CURRENTS 


71 


72 


ELECTRIC  TRANSIENTS 


Problems  and  Experiments 

1.  Let  the  sine  wave  curve  in  Fig.  55  represent  the  60  cycle  alternating 
current  that  would  flow  in  a  circuit  having  3.0  ohms  resistance,  0.05  henrys 
inductance  for  a  given  voltage. 

(a)  Let  the  switch  impressing  the  voltage  on  the  circuit  be  closed  at  the 
instant  marked  (a)  in  the  diagram.  Draw  in  rectangular  coordinates: 

1.  The  permanent  current  sine  wave  as  in  Fig.  55. 

2.  The  starting  transient. 

3.  The  actual  current  flowing  in  the  circuit  during  the  first  ^  second 
after  the  switch  is  closed. 


\ 


FIG.  55. — Single  phase  current,  sine  wave,  60  cycle  starting  transient. 

(6)  Similar  to  (a)  except  the  voltage  is  impressed  at  the  instant  marked 

(6). 

2.  In  a  circuit  having  60  ohms  resistance  and  0.045  henrys  inductance  a 
25  cycle  current  is  flowing,  as  represented  by  the  sine  wave  on  the  left  side 
in  Fig.  56.  At  the  instant  marked  (a)  the  impressed  voltage  is  suddenly 
changed  so  that  it  will  produce  a  permanent  60  cycle  current  shown  by  the 
dotted  line  sine  wave  in  the  figure. 

(a)  Draw  in  rectangular  coordinates: 

1.  The  sine  current  waves  as  in  Fig.  56. 

2.  The  starting  transient. 

3.  The  actual  60  cycle  current  for  the  first  ^lo  second  after  the  voltage 
was  changed. 

(6)  Same  as  (a),  except  the  change  is  made  at  some  other  point  along  the 
time  axis. 

3.  Take  an  oscillogram  of  the  starting  current  in  a  circuit  of  known 
resistance  and  inductance.  Calculate  the  starting  transient  and  draw  it 
on  the  oscillogram.  Check  by  combining  the  ordinates  for  the  actual  current 


ALTERNATING  CURRENTS 


73 


recorded  by  the  oscillograph  with  the  corresponding  values  of  the  calcu- 
lated transient  and  compare  the  resulting  curve  with  the  permanent  cur- 
rent sine  wave. 

4.  Let  the  sine  waves  in  Fig.  57  represent  the  permanent  value  of  the 
currents  flowing  in  a  balanced  three-phase  system,  whose  time  constant  is 
M>,ooo  °f  &  second. 


FIG.  56. — Single   phase   current,    sine   wave,    25   cycles   to   60   cycles   transient. 


FIG.  57. — Three  phase  current,  sine  wave,  60  cycle  starting  transients. 

(a)  Let  the  voltage  be  impressed  at  the  instant  marked  (a).     Draw  in 
rectangular  coordinates : 

1.  The  permanent  current  sine  waves  as  in  Fig.  57. 

2.  The  starting  transients  for  the  three  phases. 


74  ELECTRIC  TRANSIENTS 

3.  The  actual  currents  as  would  be  recorded  by  an  oscillograph 
if  a  vibrator  was  connected  to  each  of  the  three  phases  so  as  to 
record  the  current-time  curves. 

(6)  Same  as  (a),  except  the  voltage  is  impressed  at  the  instant  marked  (6). 

6.  Take  an  oscillogram  of  the  starting  currents  in  a  three-phase  system 

connecting  the  vibrators  as  in  the  circuit  diagram  in  Fig.  34.     From  the 

oscillogram  and  the  circuit  constants  plot  the  starting  transients  and  check 

with  the  permanent  current  waves  as  explained  in  Prob.  3. 

6.  Make  oscillograms  similar  to  Figs.  40  and  42  or  41  and  43.     Obtain 
the  necessary  data  to  draw  the  scale  in  amperes  or  volts  for  each  vibrator. 
A  circuit  diagram  showing  the  position  of  each  vibrator  should  be  attached 
to  each  film. 

7.  Make  oscillograms  similar  to  Figs.  46  and  48  or  47  and  49.     Quanti- 
tative data  should  be  obtained  for  each  vibrator  and  for  the  circuit  constants. 

8.  Make  an  oscillogram  similar  to  Fig.  53. 


CHAPTER  V 
DOUBLE  ENERGY  TRANSIENTS 

Single  energy  transients  occur  in  electric  circuits  or 
other  apparatus  in  which  energy  can  be  stored  in  only  one 
form.  Any  change  in  the  amount  of  energy  stored  produces 
transients  and  whether  the  stored  energy  is  decreased  or 
increased  the  transient  itself  is  a  decreasing  function  with 
its  maximum  value  at  the  first  instant.  In  magnetic, 
electric  and  dielectric  circuits  in  which  the  resistance, 
inductance  and  condensance  are  constant  during  the  transi- 
tion period,  single  energy  transients  may  be  expressed  by 
the  exponential  equation  as  discussed  in  Chaps.  Ill  and  IV. 

In  apparatus  having  two  forms  of  energy  storage  as  a 
pendulum  or  electric  circuits  having  both  inductance  and 
condensance,  a  series  of  oscillations  may  take  place  by 
which  the  energy  is  transferred  from  one  form  to  the  other, 
while  the  dissipation  of  the  stored  energy  into  heat  proceeds 
in  much  the  same  manner  as  in  single  energy  systems. 
Thus  a  pendulum,  freely  suspended  in  air,  will  swing  back 
and  forth  over  arcs  of  decreasing  amplitude,  with  energy 
changing  from  the  kinetic  to  the  potential  form  and  back 
to  the  kinetic  twice  for  each  cycle.  The  amplitude  of  each 
swing  is  less  than  for  the  one  preceding  since  part  of  the 
energy  has  been  dissipated  into  heat  by  friction  during  the 
intervening  time.  The  pendulum  comes  to  rest  when  all 
the  stored  energy  is  dissipated  into  heat. 

In  electric  circuits  having  both  dielectric  and  magnetic 
storage  facilities  the  energy  stored  in  one  form  may  change 
to  the  other  and  back  and  forth  in  a  series  of  oscillations 
of  definite  frequency.  This  is  illustrated  by  the  oscillogram 
in  Fig.  58.  The  energy  stored  in  a  condenser  is  discharged 
through  a  resistance  in  series  with  an  inductance.  In 

75 


76 


ELECTRIC  TRANSIENTS 


o  <» 

iO    ^ 


DOUBLE  ENERGY  TRANSIENTS  77 

passing  from  the  dielectric  field  to  the  magnetic  field  or 
the  reverse,  the  energy  goes  through  the  resistance  and  a 
part  is  dissipated  by  the  Ri2  losses.  Hence  the  amplitude 
of  each  oscillation  is  less  than  for  the  one  preceding.  By 
referring  to  the  timing  wave  on  the  oscillogram,  Fig.  58, 
it  is  found  that  the  frequency  of  oscillation  was  1,070  cycles 
per  second,  and  that  practically  all  the  energy  was  dissi- 
pated into  heat  by  the  Ri2  losses  in  50  cycles,  or  approxi- 
mately Ho  of  a  second. 

Surge  or  Natural  Impedance  and  Admittance. — If  no 
energy  is  dissipated  during  the  transfer  the  stored  energy  in 
the  dielectric  field  when  the  voltage  is  a  maximum  must  be 
equal  to  the  quantity  stored  in  the  magnetic  field  when  the 
current  is  a  maximum.  Hence  from  (7)  and  (16) 

Cf       Lf  (120) 

Therefore,  from  (120) 

Mj   =  ^\~  =  nz<  the  surge  or  natural  impedance  of 

the  circuit     (121) 

"I          1C 

fY,  =  A/7    —  ny,  the  surge  or  natural  admittance  of 
rj        \L 

the  circuit     (122) 

The  quantity,  \/L/VC,  is  in  the  nature  of  an  impedance 
and  is  called  the  surge  or  natural  impedance  of  the  circuit, 
and  its  reciprocal,  \/C/\/L,  the  natural  or  surge  admit- 
tance of  the  circuit. 

Frequency  of  Oscillation  in  Simple  Double  Energy 
Circuits. — Consider  circuits  "a"  and  "6"  in  Fig.  59.  Let 
the  inductance,  L,  the  condensance,  C,  and  the  resistance, 
/2,'be  constant  and  of  the  same  value  in  the  two  cases.  Let 
an  alternating  current  voltage  be  impressed  on  the  ter- 
minals and  let  the  frequency  be  varied  until  the  current  is 
in  phase  with  the  voltage  at  the  terminals.  All  the  energy 
absorbed  by  the  Ri2  losses  is  supplied  from  the  a.c.  mains. 


78 


ELECTRIC  TRANSIENTS 


In  circuit  (a)  under  the  given  conditions: 

JL%  —  jcX  =  0 


Hence, 


Likewise  for  circuit  (b) 

jcb  -  jLb  =  0 


Hence, 


(121) 

(122) 

(123) 



(124) 
(125) 

(126) 


The  expressions  in  equations  (123)  and  (126)  are  generally 
used  to  determine  the  "resonance  frequency"  of  the  cir- 


(a)  (b) 

FIG.  59. — Simple  series  and  parallel  double  energy  circuits. 

cuits.  As  shown  in  Chap.  VIII  a  strict  application  of  the 
definition  for  resonance  gives  a  different  value  for  the  true 
resonance  frequency  unless  the  resistance  is  negligible. 
If  all  the  resistance  were  removed  from  the  circuits  in 
Fig.  59  no  energy  would  be  supplied  from  the  bus  bars  and 
the  stored  energy  would  be  transferred  back  and  forth 
between  the  inductance  and  the  condensance.  With  no 
losses  the  frequency  of  the  natural  or  free  oscillations 
would  be  the  same  as  the  "  resonance  frequency"  given  in 
equations  (123)  and  (126). 


DOUBLE  ENERGY  TRANSIENTS 


79 


In  circuit  a,  Fig.  60,  and  for  the  oscillograms  in  Figs. 
61  to  65  the  condenser  is  charged  from  a  direct  current 
supply  main  after  which  the  switch  "  S"  is  thrown  to  the 
right  so  as  to  form  an  independent  closed  circuit  with  the 
condenser,  C,  resistance,  R,  and  inductance,  L,  in  series. 
The  energy  stored  in  the  condenser  is  dissipated  into  heat 
by  the  Ri2  losses  during  a  series  of  oscillations  between  the 
dielectric  and  magnetic  fields. 


FIG.  60:— Simple   oscillatory  double  energy   circuits. 

From  Kirchoff's  Laws  the  voltage  in  the  closed  circuit, 
Figs.  60  to  64,  while  the  energy  originally  stored  in  the 
condenser  is  dissipated  into  heat,  is  expressed  by  equations 

(127)  or  (128). 

=  0  (127) 


dt*  +  Rdi  +  (J  =  °  (128) 

This  is  a  homogeneous  differential  equation  of  the  second 
order  and  its  general  solution  is  given  by  equation  (129), 
in  which  A\  and  A 2  are  the  arbitrary  constants. 

(129) 
(130) 


In  equation  (129) 


B 


LC 


R 


LC 


(131) 


80 


ELECTRIC  TRANSIENTS 


FIG.  61. — Double  energy  transient. 

E  =  120  volts;  R  =  40  ohms;  G  =  0;  L  =  0.205  henrys;  C 
farads;  timing  wave  100  cycles. 


0.813    mtero- 


FIG.  62. — Double  energy  transient. 

E  =  120  volts;  R  =  75  ohms;  G  =  0;  L  =  0.205  henrys;  C   =  0.873  micro- 
farads; timing  wave  100  cycles. 


DOUBLE  ENERGY  TRANSIENTS 


81 


FIG.  63. — Double  energy  transient. 

E  =  120  volts;  R  =  150  ohms;  G  =  0;  L  =  0.205   henrys;  C  =  0.813  micro- 
farads; timing  wave  100  cycles. 


FIG.  64. — Double  energy  transient. 

E  =  700  volts;  R  =  770  ohms;  G  =  0;  L  =  0.205  henrys;  C 
farads;  timing  wave  100  cycles 
6 


0.813  micro- 


82  ELECTRIC  TRANSIENTS 

In  order  to  more  readily  keep  the  dissipation  or  damping 
factors  separate  from  the  parts  indicating  oscillations, 
equations  (130),  (131)  are  rewritten  in  (132),  (133): 

R 

Ul    =     ~nr    + 


R        .    I  I 
--3- 


From  (129),  (132),  (133): 

_  2"  •  \  orr/2"  ,        _m  _  ;\i  L  T  ~&  t 

i    =    An     2Le3\LC       4L«'+^ae      2Le      3\LC       4L«  f        (134) 

But  from  Euler's  equation  for  the  sine  and  cosine: 

••v/JL  _  *! 
\LC    4Li  =    ;"  =  ^  +  j  sin  co^  (135) 


rf  -  j  sin  ut         (136) 
Hence  from  (134),  (135),  (136): 

_Rt  _Rt 

i  =  Aie  ^[cos  ut  +' sin  ut]  +  A2e   2L 

•      •  n        (137) 

[cos  ut  —  j  sin  at] 
From  (135),  (136): 

co   ==   ^f=^-C~f^  (138) 

Hence, 

/  =  2W/!c-£  (139) 

In  circuits  for  which  the  quantity  under  the  radical  is 
real,  oscillations  occur  at  a  definite  frequency  as  determined 
by  equation  (139)  and  as  illustrated  by  the  oscillograms  in 
Figs.  61  to  63. 

If  the  resistance, 

.  R  >2  J5  (140) 


DOUBLE  ENERGY  TRANSIENTS  83 

the  quantity  under  the  radical  sign  in  (139)  becomes  imagin- 
ary, and  hence  the  circuit  is  non-oscillatory.  All  the 
energy  initially  stored  in  the  condenser  is  dissipated  into 
heat  as  the  voltage  and  current  decrease  to  zero.  This 
condition  is  illustrated  by  the  oscillograms  in  Figs.  64  and 
65. 

For  circuits  having  comparatively  little  resistance  the 
naturalfrejjuency  of  oscillations,  as  given  in  equation  (139), 
is  ver$;  Dearly  the  same  as  the  "  resonance  frequency" 
given  by  equations  (123)  or  (126).  Thus  for  the  circuit 
data  in  Fig.  62  the  nafuFal  frequency  of  oscillation,  using 
equation  (139),  is  given  in  equation  (141),  while  the  cor- 
responding "  resonance  frequency"  from  equation  (126), 
is  given  in  equation  (142). 


1      /  1 

ir\LC  " 


4/2  cycles  per  second 


f  =  n      /T  ^  =  391  cycles  per  second  (142) 


For  circuits  corresponding  to  the  conditions  that  would 
exist  if  the  condenser  in  Fig.  606  were  leaky,  similar  equa- 
tions may  be  obtained.  The  voltage  equation,  based  on 
KirchofPs  Laws  for  circuits  of  the  type  shown  in  Fig.  606, 
and  in  Figs.  66  to  70,  is  expressed  by  equations  (143)  and 
(149). 

L%  +  *$  +  c  =  °  (143) 

Using  the  notation  shown  in  the  circuit  diagram,  Fig. 
606,  letting  ce  be  the  voltage  across  the  condenser  terminals, 
and  applying  Ohm's  and  KirchofPs  Laws. 

d  =  i  +  Oi  (144) 

d  =  GLe  (145) 

dj 
ce  =  Ri'+L™  (146) 

Hence, 

di 

ci  =  i  +  GRi  +GL  (147) 


84 


ELECTRIC  TRANSIENTS 


DOUBLE  ENERGY  TRANSIENTS  85 

From  (143)  and  (147), 

LCdjl  +  (RC  +  GL)~.+  (1  +  GK)i  =  0       (148) 
at  ctt 

or, 

M'       '*•<**       '^-M  =  0      (149) 


Equation  (149)  is  a  homogeneous  differential  equation 
of  the  second  order  of  the  same  form  as  equation  (133). 
Hence,  the  same  general  solution  applies  to  both  equations, 
as  expressed  by  equation  (150),  in  which  B}  and  B2  are  the 
two  arbitrary  constants. 

+  B2ev*  (150) 


*  -  - 


Rewriting  (151),  (152)  so  as  to  more  clearly  indicate  the 
damping  and  oscillation  factors,  equations  (153),  (154)  are 
obtained. 


From  (150),  (153),  (154), 

~\(L  +  c)'€-  y  "N/Lc  -  KL  -  ?)'«       (155) 


From  Euler's  equation, 


_        _ 

C   4-      c<  =  e^  =  cos  ut  +   sin 


V_L  _  i/j?  _  o\ 
c        LC     iU     e^=  e     wf=  cos  ut  _jsin  ^        (157) 

Hence, 


[cos  cot  +  j  sin  cotj 

^2e    2^L     c/  rcos  ^1  —  j  sin 


86 


ELECTRIC  TRANSIENTS 


FIG.  66. — Double  energy  transient. 

~E  =  625  volts;  R  =  4.5  ohms;  G  =  1.67  •  10~4  mhos.;  L    =  0.205  henrys;  C 
0.813  microfarads;  timing  wave  100  ovnles. 


FIG.  67. — Double  energy  transient. 

E  =  640  volts;  R  =  4,.5  ohms;  G  =3.33  •  10"4  mhos;  L  =  0.205  henrys;  C 
0.313  microfarads;  timing  wave  100  cycles. 


DOUBLE  ENERGY  TRANSIENTS 


87 


From  (156),  (157), 


' = 2  &  -¥ 

Zir\LL        4\ 


fi 

L 


G 


(159) 


(160) 


(161) 


The  circuit  is  non-oscillatory  if 
R  _G      _2_ 
L      C  >  \LC 
For  circuits  in  which  the  quantity  under  the  radical  sign 
is  greater  than  zero,  the  energy  in  the  condenser  will  be 
dissipated  into  heat  during  a  series  of  oscillations  of  definite 
frequency  as  determined  by  equation  (160)   and  as  illus- 
trated by  the  oscillograms  in  Figs.  66,  67  and  68. 


FIG.  68. — Double  energy  transient. 

E  =  625    volts;    R  =  4.5    ohms;    G  =  6.66  •  10~4  mhos;  L  =   0.205   henrys; 
C  =  0.813  microfarads;  timing  wave  100  cycles. 

If  the  resistance  and  the  conductance  are  of  such  values 
relatively  to  the  inductance  and  the  condensance  that  the 
quantity  under  the  radical  sign  in  (160)  becomes  imaginary, 


88 


ELECTRIC  TRANSIENTS 


FIG.  69. — Double  energy  transient. 

E  =  550  volts;  R  =  4.5    ohms;   G  =  1.67  •  10~3   mhos; 
C  =  0.813  microfarads;  timing  wave  100  cycles. 


0.205    henrys; 


FIG.  70. — Double  energy  transient. 

E  =  550    volts;    R  =  4.5    ohms;   G  =  4.4  •  110~3   mhos; 
C  =  0.813  microfarads;  timing  wave  100  cycles. 


L  =  0.205    henrys; 


DOUBLE  ENERGY  TRANSIENTS  89 

the  circuit  would  be  non-oscillatory.  The  energy  initially 
stored  in  the  condenser  is  dissipated  into  heat  while  the 
voltage  and  current  decrease  to  zero,  as  illustrated  by  the 
oscillograms  in  Figs.  69  to  70. 

The  circuits  in  which  the  resistance  and  conductance 
are  comparatively  small  the  natural  frequency  of  oscilla- 
tion is  very  nearly  the  same  as  the  resonance  frequency  or 
the  natural  frequency  of  circuits  in  which  R  and  G  are 
equal  to  zero.  Thus  for  the  circuit  in  Fig.  67  the  natural 
oscillation  frequency, 


f  =  ~  :=  391  CyCl6S  PCT  S6COnd  (162) 


Considering  R  and  G  as  negligible  in  determining  the 
frequency  of  oscillation, 

/  =  -  —7—  =  391  cycles  per  second  (163) 

2ir\/LC 

A  very  interesting  circumstance  is  revealed  by  equation 
(160).  A  circuit  having  resistance  greater  than  the  critical 
value  for  oscillatory  discharges  as  given  in  (161),  may  be 
made  oscillatory  by  increasing  the  conductance  across  the 
terminals  of  the  condenser  without  changing  the  resistance. 
This  is  illustrated  by  the  oscillograms  in  Figs.  71,  72,  73 
and  74. 

For  the  given  circuit  constants  in  Fig.  71  the  circuit  is 
non-oscillatory.  Letting  R,  L  and  C  remain  constant  and 
of  the  same  value  as  in  Fig.  71  but  increasing  the  conduc- 
tance, G,  the  circuit  is  made  oscillatory  in  Fig.  72  although  the 
damping  factor  is  greater  than  for  the  circuit  in  Fig.  71. 
In  Fig.  73  the  oscillation  was  greatly  reduced  and  by  still 
further  increasing  the  conductance  while  R,  L  and  C  remain 
constant,  the  circuit  is  again  made  non-oscillatory  as  shown 
by  the  oscillogram  in  Fig.  74. 

Dissipation  Constant  and  Damping  Factor  in  Simple 
Double  Energy  Circuits.—  In  the  solution  for  the  current  in 
double  energy  circuits,  Fig.  60a  and  Figs.  61  to  65,  as 
given  in  equation  (137),  the  damping  factor  and  the  dissi- 


90 


ELECTRIC  TRANSIENTS 


FIG.  71. — Double  energy  transients. 
E  =  700   volts;    R  =  150    ohms;   G  =  0   mhos;   L  =  0.205    henry 
microfarads;  timing  wave  100  cycles. 


C  =  36 


FIG.  72. — Double  energy  transients. 

E  =  700  volts;  R  =  150 -ohms;  G  =  4.35  X  10~3  mhos;  L 
C  =  36  microfarads;  timing  wave  100  cycles. 


0.205    henry s; 


DOUBLE  ENERGY  TRANSIENTS 


91 


FIG.  73. — Double  energy  transients. 

E  =  400    volts;    R  =  150    ohms;   G  =  1.31  •  10~2   mhos;   L  =  0.205    henrys 
C  =  18  microfarads;  timing  wave  100  cycles. 


FIG.  74. — Double  energy  transients. 

E  =  700   volts;    R  =  150   ohms;   G  =  2.63  •  10~2   mhos; 
C  =  36  microfarads;  timing  wave  100  cycles. 


0.205    henrys; 


92  ELECTRIC  TRANSIENTS 

pation  constant  have  already  been  found.  Similarly 
for  circuits  in  Fig.  606  and  in  Figs.  66  and  67,  the  factors 
may  be  obtained  from  equation  (158) 

Dissipation  or  damping  constant  =  —  J(T  +  f ) 

Damping  factor  =€~2\L  +  c)i        (1(35) 

While  the  above  expressions  are  obtained  mathematic- 
ally by  the  solution  of  the  differential  equation  of  the  cir- 
cuit, it  is  important  that  the  student  gain  a  clear  concept  of 
the  physical  phenomena  involved. 

In  Chap.  Ill  it  was  shown  that  for  single  energy  tran- 
sient in  circuits  having  resistance  and  inductance  in  series 
the  time  constant  is  directly  proportional  to  the  inductance 
and  inversely  to  the  resistance. 

,T1  --  ^  (166) 

Similarly  for  circuits  having"  condensance  in  parallel 
with  conductance,  the  time  constant  is  directly  proportional 
to  the  condensance  and  inversely  to  the  conductance. 

CT,  --  g  (167) 

In  double  energy  circuits  the  energy  is  alternately  stored 
in  the  magnetic  and  dielectric  fields.  In  circuits  having 
inductance,  resistance,  condensance  and  conductance, 
arranged  as  shown  in  the  circuit  diagrams  in  Figs.  66  to  70, 
energy  is  dissipated  into  heat  both  in  the  resistance  and  in 
the  conductance.  The  rate  of  dissipation  is  greatest  in 
the  conductance,  (re2,  when  the  voltage  across  the  condenser 
is  a  maximum,  that  is,  at  the  instant  all  the  energy  is 
stored  in  the  dielectric  field.  Similarly  the  rate  of  dissipa- 
tion in  the  resistance,  Ri2,  is  a  maximum,  when  the  current 
is  a  maximum,  that  is,  when  all  the  energy  is  stored  in  the 
magnetic  field.  It  is  evident  that  since  the  energy  is 
oscillating  it  will  be  in  the  dielectric  field  half  of  the  time 
and  in  the  magnetic  field  half  of  the  time.  Since  the  energy 


DOUBLE  ENERGY  TRANSIENTS  93 

is  in  the  dielectric  form  only  half  the  actual  time,  the 
rate  of  dissipation  in  the  conductance  will  be  equal  to  half 
of  what  would  be  the  case  for  the  same  circuit  constants 
in  the  corresponding  single  energy  transient.  Hence,  the 
time  constant,  CT^  for  the  dielectric  half  of  the  double  energy 
circuit  would  be  twice  the  time  constant,  cTi,  in  the  corre- 
sponding single  energy  transient. 

9C 

J\  =  2or,  =  ^T  (168) 

(jT 

Similarly  the  time  constant,  ,7%,  for  the  inductance- 
resistance  part  of  the  double  energy  circuit  would  be  twice 
the  time  constant,  LTi,  for  the  corresponding  single  energy 
transient. 

or 

,T2  =  2LT1  =  -£  (169) 

K 

Under  the  given  circuit  conditions  with  R,  L,  G  and  C 
constant,  the  proportionality  law  applies  to  double  energy 
transients  on  the  same  basis  as  for  single  energy  transients. 
The  transient  term  is  therefore  expressed  by  the  exponential 
equation  and  appears  as  a  factor  in  the  current-time  and 
voltage-time  equations  and  represents  the  dissipation  of 
energy  into  heat  by  the  resistance,  Ri2,  and  the  conduc- 
tance, Ce2,  in  the  circuit. 

Let  u  represent  the  dissipation  constant  of  double  energy 
circuits.  The  damping  factor  is  therefore, 

1        l          -Rt  -°-t 

ut    =  ,T>        CT2    =  2L         2G 


c  (17!) 

\/R       .       G\  /ihr<n\ 

-2(1  +  c)  (172) 

This  is  the  same  value  as  obtained  in  equation  (158). 
For  circuits  in  which  G  =  0,  as  illustrated  by  Figs.  61 
to  65,  the  term  G/C  would  be  zero. 


94  ELECTRIC  TRANSIENTS 

Hence, 


u'       -g-  (173) 

-  Rt 
eu't    =    e      2L  Q74) 

This  corresponds  to  the  value  of  the  damping  factor  in 
equation  (137). 

For  circuits  similar  to  Figs.  66  to  70  but  in  which  R  =  0, 
the  term  R/L  would  be  zero. 

Hence, 

u"  =  -£c  (175) 

_G_t 

eu"t  =  €   2c  (175) 

As  it  is  not  possible  to  completely  eliminate  the  resis- 
tance in  circuits  having  inductance,  the  conditions  for  u" 
can  not  be  fully  realized  experimentally. 

Equations  for  Current  and  Voltage  Transients. — For 
simple  double  energy  circuits,  with  R,  L  and  C  in  series, 
as  in  Fig.  60a,  the  general  equation  (137)  for  the  current  is, 

_Rt 

i  =  Aie   2Z/[cos  cot  +  j  sin  cofl 

_  Rt 

+  A2e~2L  [cos  cot  —  j  sin  co(]     (177) 

In  equation  (177)  A\  and  A2  are  the  arbitrary  constants, 
which  for  any  specific  case  are  determined  by  the  perman- 
ent circuit  conditions  preceding  and  following  the  transient 
period.  Equation  (177)  may  be  written  in  a  more  compact 
form  as  in  (178),  in  which  A3  and  A  4  are  the  arbitrary 
constants  which  for  any  specific  case  may  be  determined 
from  the  given  limiting  conditions  under  which  the  tran- 
sient occurred. 

_Rt 

i  =  e    2L[A3  cos  ut  +  A±  sin  co£]  (178) 

The  voltage  across  the  terminals  of  the  condenser, 

ce  =  -Ri  -  L  (179) 


DOUBLE  ENERGY  TRANSIENTS  95 

From  (178),  (179), 
ce  =  e   2L  I  rt~[A3  cos  co£  +  A  4  sin  co£] 

+  wL[A4cosco^  —  A3sincofl        (180) 

For  the  transients  in  Figs.  61  to  65  the  starting  conditions 
are: 

t  =  0;  i  =  0;  ce  ==  #  (181) 

From  (178),  (180),  (181), 

Hence, 

p  -*1 

L  sin  w<  (183) 


CO-L 


Rt  7? 

2L[cos  co^  +  0^  sin  co£]  (184) 


T> 

For  the  given  circuit  constants,  ^  j  is  very  small  and 
hence, 

_Rt 

ce  =  Ee    2L  sin  ut  (very  nearly)  (185) 

To  illustrate  the  application  of  equations  (183)  and  (185) 

for  the  solution  of  numerical  problems,  equations   (186) 

and  (187)  give  the  value  of  the  current  and  voltage  in 

amperes  and  volts  for  the  oscillograms  in  Fig.  62. 

i  =    -0.24  €~182"sin  (170760°*)  amperes  (186) 

e  =-.  120.  e~182'cos  (170760°0  volts  (187) 

For  the  circuit  in  Fig.  606  the  equations  are  of  a  similar 
nature.  The  general  equation  (158)  for  the  current  is 
given  in  (188)  and  may  be  written  in  a  more  compact 
form  as  in  equation  (189),  in  which  Bs  and  B  4  are  constants 
that  in  each  case  depend  on  the  permanent  circuit  condi- 
tions preceding  and  following  the  transient  period. 


[cos  coZ  +  j  sin  <*t] 


2L     c 


[cos  ut  -  j-sin.ut]     (188) 


96  ELECTRIC  TRANSIENTS 


-  t 

i  =  €    2\L     c>  [B3  cos  ut  +  B4  sin  <*t]  (189) 


ce  =  -Ri  _  L--  (190) 


di 
-- 

From  (188)  and  (190), 

cos  ut  +  ^4  sin 


+  <o£[/?4  cos  cot  —  J53  sin  oo£l      (191) 

For  the  transients  in  Figs.  66  to  70  the  initial  conditions 
are: 

t  =  0;t  =  0;  ce  =  E  (192) 

From  (189),  (191)  and  (192), 

£3  =  0;  B,  =         Jr  (193) 

Hence, 
i=-  4«  '^*  +  ^'sin««  (194) 

(195) 


If  in  equation  (195), 


_  , 

ce  =  Ee    2\L  +  c)  cos  ut  (197) 

and 

(198) 


As  an  illustration  of  the  application  of  equations  (194) 
and  (197)  to  the  solution  of  a  specific  problem,  let  the 
numerical  values  of  the  circuit  constants  in  Fig.  67  be 
used.  Equations  (199)  and  (200)  give  the  instantaneous 
values  of  the  current  and  voltage  for  the  oscillogram  in 
Fig.  67. 

_  9  1  r,  i 

i=-l.28e'     sin  (170760°*)  amperes         (199) 


ce  =  640  e'       cos  (170760°0  volts  (200) 


DOUBLE  ENERGY  TRANSIENTS 


97 


o 

o 

LO^ 

II  J 
^'E 

fi 


O    c3 

2  3 


o 
w>  S 

03 


98 


ELECTRIC  TRANSIENTS 


FIG.  76. — Starting  current  and  voltage  transients. 

E  =  125  volts;  R  =  5.0  ohms;(?  =  0.00167  mhos;L  =  0.205  henrys;  C  =  9.0 
microfarads;  timing  wave  100  cycles. 


FIG.  77.- — Starting  current  and  voltage  transients. 

E  =  125  volts;  R  =  5.0  ohms;  G  =  0.0025  mhos;  L  =  0.205  henrys;  C 
microfarads;  timing  wave  100  cycles. 


9.0 


DOUBLE  ENERGY  TRANSIENTS 


99 


FIG.  78. — Starting  current  and  voltage  transients. 

E  =  125    volts;    R  =  5.0  ohms;  G  =  0.005  mhos;L  =  0.205  henrys;  C 
microfarads;  timing  wave  100  cycles. 


FIG.  79. — Starting  current  and  voltage  transients. 

E  =  125  volts;  R  =  5.0  ohms;  G  =  0.0132  mhos;  L  =  0.205  henrys;  C  =  9.0 
microfarads;  timing  wave  100  cycles. 


100  ELECTRIC  TRANSIENTS 

The  oscillogram  in  Fig.  74  shows  the  current  and  voltage 
transients  in  a  circuit  having  a  high  damping  factor  but 
in  which  the  frequency  of  oscillation  is  the  same  as  if  both 
R  and  G  wrere  zero.  The  data  in  Fig.  74  show  that  the 
circuit  constants  were  of  such  values  as  to  satisfy  equation 
(196). 

For  the  oscillograms  in  Figs.  75  to  79  the  circuits  are  of 
the  same  type  as  in  Figs.  66  to  70,  but  the  permanent  con- 
ditions preceding  and  following  the  transition  period  are 
different.  The  oscillograms  show  the  starting  current  and 
voltage  transients  at  the  points  in  the  circuit  indicated  by 
the  positions  of  the  vibrators  in  the  circuit  diagram  and  for 
the  values  of  R,  L,  G  and  C,  as  given  in  each  figure. 

Problems  and  Experiments 

1.  Given  a  circuit  similar  to  Fig.  60 (a)  having  R,  L,  and  C  in  series.     Let 
R  =  20  ohms,  L  =  0.31  henrys,  C  =  1.2  microfarads  and  E  =  120  volts, 
the  initial  condenser  discharge  voltage. 

(a)  Find  the  natural  period  of  oscillation  of  the  circuit. 

(6)   Find  the  time  constant,  and  the  damping  factors. 

(c)    Write  the  equation  for  the  transient  condenser  discharge  current. 

(d}  For  what  values  of  R  would  the  circuit  be  non-oscillatory. 

2.  Derive  the  equations  for  ee,  the  transient  voltage  across  the  condenser 
terminals  in  Fig.  62.     Trace  the  voltage-time  curve  for  ce  on  rectangular 
coordinates,  using  the  same  time  scale  on  the  X  axis  as  in  the  oscillogram. 

3.  Take  a  double  energy  oscillogram  similar  to  Fig.  58.     Obtain  all  the 
necessary  data  and  write  the  equations  for  the  transient  current. 

4.  Write  the  equations  for  the  voltage  and  current  curves  of  the  oscillo- 
gram in  Fig.  61  similar  to  equations  (199)  and  (200)  for  Fig.  67  in  the  text. 

5.  Take  a  series  of  oscillograms  similar  to  Figs.  66  to  70.     Find  the  values 
of  the  circuit  constants  and  place  on  the  film  ampere  and  volt  scales  for  the 
current  and  voltage  curves. 

6.  For  the  oscillogram  in  Fig.  75  with  the  given  circuit  conditions: 
(a)   Write  the  expression  for  ce  and  i  similar  to  equations  (194),  (195). 
(6)   Insert  the  numerical  values  of  circuit  constants  and  express  ce  and 

i  in  volts  and  amperes,  similar  to  equations  (199)  and  (200). 


CHAPTER  VI 

ELECTRIC  LINE  OSCILLATIONS,  SURGES  AND 
TRAVELING  WAVES 


Electric  lines  whether  designed  for  poW«er*tVaVL&fnissioTi  or 
telephone  service,  may  be  considered  "as^cpnisisjifig^f  8it 
infinite  series  of  infinitesimal  double  energy  'Circuits  o*f  the 
simple  types  discussed  in  Chap.  V.  Each  infinitesimal 
length  of  line  may  be  represented  by  the  resistance  and 
inductance  in  one  of  the  series  circuit  elements  in  Fig.  80 
and  the  corresponding  portion  of  the  dielectric  between 
the  conductor  and  neutral  by  the  conductance  and  con- 
densance  in  the  adjacent  parallel  circuit.  The  line  con- 
stants, R,  L,  G  and  (7,  depend  on  the  size  and  spacing  of  the 
conductors  and  the  electrical  properties  of  the  dielectric 
and  conductor  materials.  To  readily  gain  clear  concepts 
of  transmission  line  phenomena  it  is  essential  for  the  student 
to  conduct  experiments  and  obtain  quantitative  test  data. 
Commercial  transmission  lines  are  seldom  available  for 
experimental  work  but  artificial  lines  having  the  electrical 
characteristics  of  actual  lines  can  be  readily  constructed 
of  convenient  design  for  operation  in  the  laboratory. 

Artificial  Electric  Lines.— Since  the  operating  charac- 
teristics of  transmission  lines  are  determined  by  the  line 
constants,  the  resistance,  inductance,  conductance  and 
condensance  and  are  independent  of  the  space  and  mass 
factors,  much  of  the  experimental  work  can  to  good  advan- 
tage be  performed  on  equivalent  artificial  electric  lines.1 
The  oscillograms  of  electric  line  transients  used  for  illus- 
trations in  this  chapter  were  obtained  from  an  artificial 
transmission  line,2  one  section  of  which  is  shown  in  Fig. 

1  DR.  A.  E.  KENNELLY,  "Artificial  Transmission  Lines." 

2  Trans.  A.  I.  E.  E.,  vol.  31,  p.  1137. 

101 


102 


ELECTRIC  TRANSIENTS 


81.  This  line  is  of  the  lumpy  "T"  type  of  design,  which 
means  that  each  unit  has  resistance  and  inductance  in 
series  combined  with  condensance  and  conductance  in 
parallel  as  shown  in  Fig.  82.  If  the  insulation  is  sufficiently 
high  the  conductance  factor  may  be  omitted  and  the  section 
circuit  diagram  would  be  as  in  Fig.  83,  which  represents 
the  circuit  diagram  for  the  "T"  unit  in  Fig.  81. 


R  L 


C  =tr^G    C=t^G     C=±= 


R  L        L  R 


R  L 


FIG.  80. — Transmission  line  circuit  diagram  showing  three  elements. 

In  the  lumpy  types  the  line  constants  R,  L,  G  and  C,  are 
massed  instead  of  uniformly  distributed  as  in  actual  lines. 
As  the  lumpy  type  only  approximates  a  uniform  distribu- 
tion of  the  resistance,  inductance,  conductance  and  con- 
densance in  the  line,  the  size  of  each  unit  must  be  small  in 
comparison  to  the  total  length  of  the  line.  In  Fig.  81 
is  shown  one  of  the  twenty  ten-mile  units  in  the  artificial 
transmission  line  in  the  electrical  engineering  laboratories 
of  the  University  of  Washington.  In  each  unit  the  line  con- 
stants may  be  adjusted  within  the  following  limits: 

Resistance,  minimum  value,  2.59  ohms. 

Inductance,  maximum  value,  0.021  henry. 

Condensance,  0.1  to  1.0  microfarad. 

The  resistance  may  be  increased  to  any  desired  amount  by 
moving  the  clamp  on  the  resistance  loop  or  by  inserting 
resistance  elements  between  the  units;  the  inductance 
may  be  decreased  by  turning  the  right  hand  coil  and  by 
taps  in  the  lower  coil;  and  the  condensance  may  be  varied 
in  steps  by  using  ten  or  a  less  number  of  condensers  in 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      103 


series.  Adjustments  can  be  made  so  as  to  give  to  the  artifi- 
cial line  the  electrical  constants  equivalent  to  an  actual 
transmission  line  of  any  size  of  wire  up  to  No.  0000  A.W.G. 
hard-drawn  copper  and  for  any  spacing  up  to  120  inches. 


FIG.  81. — Section  of  artificial  electric  line,  University  of  Washington. 

The  line  may  also  be  adjusted  so  as  to  be  equivalent  to 
commercial  telephone  lines. 

Time,  Space  and  Phase  Angles. — In  Chap.  V  the  equations 
for  the  current  and  voltage  transients  were  derived  for 


104 


ELECTRIC  TRANSIENTS 


simple  double  energy  circuits,  Fig.  60,  in  which  the  circuit 
constants,  R,  L,  G  and  C  are  massed.  Evidently  the 
energy  transfer  between  the  magnetic  and  dielectric  fields 
would  be  of  essentially  the  same  nature  if  the  inductance 
and  resistance  were  intermixed  with  the  condensance  and 
conductance  or  uniformly  distributed  as  in  a  transmission 
line.  However,  one  important  difference  must  be  noted 
which  necessitates  an  additional  factor  in  the  expression 
for  the  transient  current  and  voltage.  In  circuits  having 
massed  circuit  constants  the  maximum  value  of  the  voltage 


f=f      /? 


FIG.  82. — T-circuit  with  leaky  condenser. 


FIG.  83. — T-circuit. 


will  be  impressed  on  all  of  the  condensance  at  the  same 
instant,  and  all  parts  of  the  magnetic  field  reach  a  maximum 
at  the  instant  the  current  is  a  maximum.  On  the  other 
hand,  with  R,  L,  G  and  C  distributed,  as  in  long  transmission 
lines,  the  time  required  for  the  electric  wave  to  travel  along 
the  length  of  the  line  enters  into  the  problem.  If  a  constant 
direct  current  voltage  is  impressed  at  one  end  of  an  electric 
line  a  short  but  definite  time  will  elapse  before  the  voltage 
reaches  the  other  end  of  the  line.  If  an  alternating  current 
is  transmitted  over  the  line  the  successive  waves  travel  over 
the  line  at  definite  velocity  in  the  same  manner  as  the 
impulse  from  the  direct  current  voltage.  The  maximum 
point  of  any  wave  travels  at  a  definite  velocity  as  deter- 
mined by  the  distribution  of  the  resistance,  inductance, 
conductance  and  condensance  in  the  line.  In  trans- 
mission lines  with  air  as  the  dielectric  and  with  copper  or 
aluminum  conductors  the  speed  at  which  a  wave  or  impulse 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      105 

travels  is  approximately  the  same  as  the  velocity  of  propa- 
gation of  an  electromagnetic  wave  in  space  or  the  velocity  of 
light. 

v  =  3-1010  cm.  per  second  (205) 

In  a  medium  having  a  permeability  ^  and  a  permittivity  /c, 

3-1010 

v'  =  -—  -=,-  cm.  per  second  (206) 

v  M* 

The  time  required  for  the  voltage  wave  to  travel  a 
distance  x  along  the  line  having  distributed  line  constants, 
depends  on  the  distance  and  velocity  of  propagation. 


(207) 


In  comparing  the  transient  voltage  and  current  conditions 
at  any  two  points  on  an  electric  line,  x  distance  apart,  con- 
sideration must  be  given  to  the  time  required  for  the 
propagation  of  the  electric  wave  over  the  given  distance 
and  hence  the  factor  t,  must  be  included  in  the  equations. 
In  double  energy  circuits  having  massed  R,  L,  G  and  C, 
as  in  the  oscillograms  in  Figs.  66  to  69,  and  for  oscillations 
produced  by  the  discharge  of  energy  initially  stored  in  the 
condensers,  the  instantaneous  values  of  the  voltage  and 
current,  under  the  stated  conditions,  are  given  in  equations 
(194),  (197).  Under  similar  conditions,  as  illustrated  by 
the  oscillograms  in  Figs.  84  to  91,  and  by  the  introduction 
of  space  angles,  the  equations  may  be  considered  as  apply- 
ing to  circuits  having  distributed  R,  L,  G  and  C,  as  in  trans- 
mission lines. 

To  simplify  the  notations,  let 


»  =  K?  -  D 


(208) 

I  =  -E  (209) 

co-L 

y  =  time  phase  angle  for  t  =  0  (210) 

For  oscillations  in  circuits  with  massed  R,  L,  G  and  C, 
under  the  stated  assumptions: 


106 


ELECTRIC  TRANSIENTS 


e  =  Ee~ut  cos  (ut  —  7) 


(211) 
(212) 


FIG.  84. — Electric  line  oscillations. 

E  =  500  volts;  R  =  52.14  ohms;  G  =  0;  L  =  0.427  henrys;  C  =  3.66  micro- 
farads; length  =  232  miles;  timing  wave  100  cycles. 

&V 


1C) 

FIG.  85. — Circuit  and  wave  diagram  for  Fig.  84, 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES       107 

For  oscillations  in  circuit  with  distributed  R,  L,  G  and  C, 
under  similar  conditions: 


FIG.  86. — Electric  line  oscillations. 

E  =  500  volts;  R  =  26.12  ohms;  #2  =  26.02  ohms;  Gi  =  0;  Gz  =  0;  Li  =  0.2128 
henrys;  Li  =  0.2146  henrys;  Ci  =  1.831  microfarads;  Cz  —  1.834  microfarads; 
timing  wave  100  cycles. 


FIG.  87. — Circuit  and  wave  diagram  for  Fig.  86.       ~  j 

i  =  Ie-ut  sin  [w(t  _  tl)  -  y]  (213) 

e  =  Ee~ut  cos  [u(t  -  ti)  --  y]  (214) 


108 

Substituting 


ELECTRIC  TRANSIENTS 

c  for  co£i : 


=     e-ut  sin  (o>£  --  0x  —  7)  (215) 

e  =  Ee~ut  cos  (ut  -  <$>x  -  7)  (216) 

In  equations  (215),  (216)  ut  is  the  time  angle,    <f)X  the 
space  angle  and  7  the  phase  angle. 


FIG.  88. — Electric  line  oscillations. 

E  =  500  volts;  Ri  =  31.28  ohms;  R2  =  15.64  ohms;  Gi  =  0;  G-i  =  0;  Li  = 
0.2564  henrys;  L2  =  0.1282  henrys;  C\  =  2.201  microfarads;  C2  =  1.10  micro- 
farads; timing  wave  100  cycles. 


FIG.  89. — Circuit  and  wave  diagram  for  Fig.  88. 

Natural  Period  of  Oscillation. — Since  the  space  angle, 
4>x,  in  equations  (215),   (216)  is  directly  proportional  to 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES       109 

the  distance,  x,  from  the  origin,  it  is  evident  that  the  phase 
of  the  current,  i,  and  the  voltage,  e,  changes  progressively 
along  the  line.  At  some  distance,  1Q,  the  current  and  volt- 


FIG.  90. — Electric  line  oscillations. 

E  =  500  volts;  Ri  =  39.10  ohms;  #2  =  13.04  ohms;  Gi  =  0;  Gz  =  0;  Li  = 
0.3204  henrys;  Z/2  =  0.1070  henrys;  Ci  =  2.748  microfarads;  (72  =  0.917  micro- 
farads; timing  wave  100  cycles. 

VBWWtfWtf^^ 


FIG.  91. — Circuit  and  wave  diagram  for  Fig.  90. 

age  are  displaced  360  deg.  from  their  starting  point  values. 
The  distance,  10,  is  called  the  wave  length  and  is  the  distance 


110  ELECTRIC  TRANSIENTS 

the  electric  field  travels  during  the  time,  tQ,  required  for  the 
completion  of  one  cycle  or  complete  wave. 

If /is  the  frequency  of  oscillations  in  cycles  per  second, 

tQ  =  -  seconds  (217) 

J 

h  =  vto  (218) 

The  fundamental  frequency  or  natural  period  of  free 
oscillation  depends  on  the  length  of  the  line  and  on  the 
imposed  circuit  conditions.  For  the  oscillations  recorded 
in  the  oscillogram  in  Fig.  84,  the  line  is  open  at  the  receiver 
end  and  connected  through  the  vibrator  circuit  at  the 
generator  end.  The  diagram  in  Fig.  856  shows  that 
under  these  conditions  the  length  of  the  line  is  one-fourth 
wave  length  of  the  fundamental  oscillations.  In  Fig.  85c  is 
shown  the  wave  diagram  for  the  ninth  harmonic  which 
appears  as  ripples  on  the  fundamental  oscillation. 

In  Fig.  86  the  vibrator  is  connected  at  the  middle  point 
leaving  both  ends  open.  The  corresponding  wave  diagram 
in  Fig.  876  shows  that  the  length  of  the  line  is  two  quarter- 
wave  lengths  or  one-half  wave  length,  and  the  frequency  of 
the  fundamental  oscillation  is  twice  that  in  Fig.  84.  Simi- 
larly in  Fig.  88,  in  which  the  vibrator  is  connected  at  one- 
third  the  distance  from  one  end,  each  of  the  two  parts 
becomes  a  vibrating  element  giving  fundamental  oscilla- 
tions. The  frequency  of  the  oscillation  of  the  shorter 
part  is  twice  as  great  as  for  the  longer  portion.  In  Fig.  90, 
with  the  vibrator  connected  at  one-fourth  the  distance  from 
one  end  of  the  line,  the  short  end  oscillates  at  three  times 
the  frequency  of  the  long  end.  In  all  cases  the  voltage 
and  current  vary  progressively  along  the  line  so  that  at 
any  instant  the  average  voltage,  instead  of  the  maximum 
value,  is  impressed  on  the  condensers  and  the  average 
current,  in  place  of  the  maximum  value,  flows  through  the 
inductance. 

The  same  results  would  be  obtained  in  circuits  with 
massed  R,  L,  G  and  C  in  which  the  maximum  voltage  is 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      111 

impressed  on  all  the  condensance  simultaneously  or  all  of 
the  magnetic  field  reaches  a  maximum  at  the  instant  the 
current  is  a  maximum,  by  reducing  the  condensance  and 
inductance  in  the  ratio  of  the  maximum  to  the  average 
values.  This  ratio  is  ir/2  for  sine  waves. 

The  frequency  for  free  oscillations  in  simple  circuits 
with  massed  R,  L,  G  and  C  was  derived  in  Chap.  V,  equation 
(162). 

~     2  cycles  per  second      (219) 


The  frequency  of  free  oscillations  in  circuits  having 
distributed  R,  L,  G  and  C  and  a  sine  wave  distribution  of 
the  voltage  and  current  may  be  obtained  by  multiplying 
L  and  C  in  equation  (219)  by  Tr/2,  the  ratio  of  the  maximum 
to  the  average  value. 


~  c    cycles  per  second    (220) 

In  commercial  electric  lines  the  quantity 


negligibly  small  in  comparison  with  1/LC.  For  practical 
problems  the  frequency  of  the  fundamental  oscillations  or 
surges  in  transmission  lines  with  uniformly  distributed 
R,  L,  G  and  C  may  therefore  be  obtained  by  equation  (221). 

/  =   .    .  n  cycles  per  second  (221) 


Thus  the  fundamental  frequency  of  oscillation  for  the 
transmission  line  in  Fig.  84, 

ThuS=  /=  -  2  0°-°  CydeS  Per  S6COnd  (222) 


This  may  be  checked  by  measurements  on  the  oscillogram 
in  Fig.  84.  On  the  original  film  (the  cut  in  the  text  is 
reduced  in  size)  10  cycles  of  the  timing  wave  measured 
14.3  cm.,  while  10  cycles  of  the  transient  oscillations 
measured  7.1  cm.  Hence  the  frequency, 

/  =  -j4^  ==  200.1  cycles  per  second  (223) 


112  ELECTRIC  TRANSIENTS 

Since  L  and  C  represent  the  total  inductance  and  conden- 
sance  of  the  line  the  frequency  depends  on  the  total  length 
of  the  line  or  the  length  of  time  in  which  the  oscillation 
occurs,  as  illustrated  by  the  oscillogram  in  Figs.  84,  86,  88 
and  90.  The  transmission  line,  or  other  circuits  of  dis- 
tributed R,  L,  G  and  C,  therefore,  have  a  fundamental 
frequency  at  which  the  whole  line  oscillates,  but  as  any 
fractional  part  of  the  line  may  also  oscillate  independently 
of  the  whole  line,  particularly  if  the  oscillating  section  is 
short  as  compared  to  the  entire  line,  oscillations  of  any 
frequency  may  occur.  At  high  frequencies  the  successive 
waves  are  so  close  together  that  a  small  variation  in  the 
time  constants  will  cause  them  to  overlap.  Since  R,  L, 
G  and  C  are  not  perfectly  constant  high  frequency  oscilla- 
tions interfere  with  each  other,  and  on  this  account  reso- 
nance phenomena  occur  only  at  low  or  moderate  frequencies. 

Length  of  Line. — In  ordinary  transmission  lines,  with  air 
as  the  dielectric  and  conductors  of  copper  or  aluminum,  an 
electric  wave  or  impulse  travels  approximately  3  1010  cm. 
per  second,  the  velocity  of  propagation  of  an  electromag- 
netic field  in  free  space,  equation  (205).  This  fact  is  of 
much  practical  importance  in  transmission  line  calculations. 
If  the  length  of  the  line  is  known  the  frequency  of  the 
fundamental  oscillation  and  of  the  harmonics  can  readily 
be  determined.  The  length  of  the  line  is  one  quarter  wave 
length  of  the  fundamental  oscillation  as  illustrated  by  the 
oscillogram  in  Fig.  84  and  corresponding  diagrams  in  Fig. 
85. 

v  =-  Wo  (224) 

Conversely,  if  the  frequency  of  the  oscillation  is  known 
the  length  of  the  oscillating  section  may  be  determined. 
In  artificial  transmission  lines  with  the  frequency  of  the 
fundamental  oscillation  obtained  from  oscillograms  the 
equivalent  length  of  the  line  can  be  calculated.  Thus  from 
measurements  on  the  oscillogram  in  Fig.  84,  equation  (223), 
/  =  200  cycles  per  second.  Hence  the  length  of  the  line, 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      113 

v       3-1010 
/o  =  4  ,-  =  cm.  =  375  km.  ==  233  miles     (225) 


From  equations  (205),  (221)  and  (224),  relations  are 
obtained  by  which  L  or  C  may  be  calculated  if  the  length 
of  the  line,  I  in  cm.,  and  either  C  or  L  are  known. 

v  =  3-10"  -  4/7  =       ~  (226) 

Hence, 


For  cables  or  circuits  in  which  the  permeability,  /*,  and 
the  permittivity,  K,  are  greater  than  unity  the  corresponding 
relations  are  obtained  from  equations  (206),  (221)  and 
(224). 

3-1010  I 


These  equations  are  useful  in  the  calculation  of  the  con- 
densance  of  circuits  in  which  the  inductance  can  be  more 
easily  determined,  as  in  complex  overhead  systems  and  in 
calculating  inductance  in  cables  or  other  circuits  in  which 
the  condensance  may  readily  be  measured. 

Velocity  Unit  of  Length.  Surge  Impedance.  —  In  hand- 
books and  tables  the  values  of  R,  L,  G  and  C  are  given  for 
some  unit  of  length  as  cm.,  km.,  1,000  ft.,  mile,  etc.  In 
discussions  and  calculations  of  transient  phenomena  the 
velocity  unit  of  length  is  sometimes  used.  For  overhead 
structures  the  unit  of  length,  I,  on  this  basis  would  be  vt  or 
3-1010  cm.  Hence  from  equation  (227),  and  under  the 
assumptions  made  in  its  derivation, 

L,  =  ~  (230) 

Ov 

The  natural  or  surge  impedance  from  equations  (121), 
(230)  : 


114  ELECTRIC  TRANSIENTS 


=  J£'-=L.  ==V  (231) 

\  U  v  ^v 


By  the  use  of  the  velocity  unit  of  length  investigations 
on  transmission  systems  having  sections  of  different  con- 
stants and  hence  of  different  wave  length  are  greatly 
simplified.  In  systems  having  overhead  lines,  cables, 
coiled  windings,  as  in  transformers,  arresters,  etc.,  the 
wave  length  becomes  the  same  in  the  velocity  measure  of 
length. 

Voltage  and  Current  Oscillations  and  Power  Surges.— 
It  has  been  shown  that  in  free  or  stationary  oscillation 
transmission  lines  or  other  electric  circuits  having  uniformly 
distributed  R,  L,  G  and  C  the  current  and  voltage  are  essen- 
tially in  time  quadrature.  From  equations  (215),  (216) : 

i  =  It-ut  sin  (w$  __  0X  __  T)  (234) 

e  =  Ee~ut  cos  (coZ  -  4>x  -  7)  (235) 

Hence,  the  instantaneous  power,  p,  at  any  point  in  the 
circuit  is  given  by  equation  (236) : 

TjJT 

p  =  ei  =  -  -  e~ut  sin  2(«J  -  <f>x  -  7)  (236) 

tU 

The  direction  of  the  flow  of  power  changes  4/  times  each 
second  since  the  sine  function  becomes  alternately  positive 
and  negative  for  successive  r  time  degrees.  That  is,  a 
surge  of  power  occurs  in  the  circuit  of  double  the  frequency 
of  the  current  or  voltage  oscillations,  although  the  average 
flow  of  power  along  the  line  is  zero. 

Average  power,  pQ  =  0  (237) 

General  Transmission  Line  Equations. — In  the  preceding 
paragraphs  various  phases  of  the  electric  transients  that 
occur  during  the  free  or  natural  oscillations  of  electric 
circuits  have  been  discussed.  The  general  problem,  in  which 
transient  phenomena  occur  while  continuous  power  is 
supplied  to  the  system  and  transmitted  along  the  line,  is 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      115 

necessarily  much  more  complex.  In  transmission  lines 
or  other  electric  circuits  having  uniformly  distributed  resis- 
tance, inductance,  conductance  and  condensance,  with 
R,  L,  G  and  C  the  constants  per  unit  length  of  line,  the 
voltage  and  current  relations  in  time  may  be  expressed  by 
partial  differential  equations  as  in  (238),  (239): 


-<       +  ft  (239) 

Differentiating  (238)  with  respect  to  x  and  (239)  with 

d^i 

respect  to  /  and  eliminating  —        equations  (240),  (241) 

(7  JU(J  L 

may  be  derived: 

^2P  x2p  3p 

LC  +  (RC  +  GL)  +  BGe     (240) 


+  (RC  +  GL)  |  +  RGi  (241) 

A  general  solution  for  these  equations  is  given  in  equation 
(242)  ,  one  term  of  which  represents  the  sum  of  the  outgoing 
and  the  other  the  sum  of  the  incoming  waves. 

e  =  Aie±at  eb*  sin  (at  +  0x  +  71) 

+  A2e±ate-b*  sin  (at  +  $x  +  T2)      (242) 

In  order  to  determine  the  values  of  A\t  A2,  a,  6,  a,  |8,  71, 
and  72,  the  specific  conditions  under  which  the  line  operates 
must  be  given.  It  is,  however,  of  first  importance  to 
understand  the  purpose  or  functions  of  each  term  in  the 
equation.  On  the  basis  of  energy  flow  and  dissipation  in 
a  line  transmitting  power  the  following  interpretation  of 
the  symbols  in  equation  (242)  may  be  helpful. 

A  i,  and  A2  are  constants  whose  values  are  determined  by 
the  limiting  conditions  of  each  specific  problem. 

e~at  may  be  called  the  time  damping  factor  and  a  the  time 
dissipation  constant  for  the  transient  oscillations. 


116  ELECTRIC  TRANSIENTS 

This  factor  represents  the  same  form  of  energy 
dissipation  as  e~wi  in  Chap.  V.  Ordinarily  the  trans- 
formation of  electric  energy  into  heat  by  the  Ri2  and 
Ge2  losses  is  non-reversible  and  therefore  the  sign  of 
the  dissipation  constant  must  be  negative. 
e ±  bx  may  be  called  the  distance  damping  factor  and  b  the 
distance  dissipation  constant.  It  relates  both  to  the 
losses  along  the  line  in  the  steady  flow  of  energy,  as  in 
transmission  lines  carrying  permanent  load,  and  to  the 
flow  of  transient  energy  in  the  system  as  with  travel- 
ing waves  or  in  the  oscillations  of  compound  circuits. 
at  is  the  time  angle.  Under  permanent  or  steady  condi- 
tions with  a  simple  sine  voltage, M  E  sin  ut,  impressed 
at  the  generating  station  a  =  w  and  has  only  one 
value.  However,  if  the  impressed  voltage  is  a 
complex  wave  or  during  transition  periods  between 
two  permanent  conditions  while  transient  currents 
and  voltages  are  flowing  in  the  system,  a  would  have 
more  than  one  value. 

fix  is  the  space  or  distance  angle  with  x  as  the  distance 
along  the  line  from  the  origin.     If  waves  of  more 
than   one  frequency   are  passing  over  the  line   |8 
would  have  more  than  one  value. 
71  and  72  are  phase  angles  for  t  =  0. 
Traveling  Waves. — Traveling  waves  are  in  many  respects 
similar  to  free  oscillations  or  standing  waves  as  the  transfer 
of  energy  between  the  dielectric  and  magnetic  fields  is  the 
basis  for  all  double  energy  electric  phenomena.     The  essen- 
tial difference  is  that  in  traveling  waves  power  flows  along 
the  line  while  in  free  oscillations  or  standing  waves  the 
energy  oscillates  between  the  two  fields  but  does  not  travel 
from  one  line  element  to  another.     Oscillograms  of  the  cur- 
rent and  voltage  factors  in  traveling  waves  are  shown  in 
Figs.   92  to  97.     It  should  be  noted  that  the  current  is 
in   time   phase  with  the  voltage  for  the  outgoing  waves 
and  differs  by  180  deg.  for  the  returning  waves.     In  both 
cases  a  flow  of  power  occurs  along  the  line. 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      117 

In  Fig.  92  the  receiver  end  of  the  line  is  short  circuited. 
The  reflected  voltage  wave  is  in  opposite  time  phase  to  the 
outgoing  wave  while  the  corresponding  current  waves  are 
in  the  same  direction. 

In  Fig.  93  the  receiver  end  of  the  line  is  open  and  as  a 
consequence  the  reflected  current  wave  reverses  in  sign 
while  the  corresponding  voltage  wave  is  in  the  same  direc- 
tion as  the  outgoing  wave. 


FIG.  92. — Traveling  waves  on  artificial  transmission  line.     Receiver  end  short 

circuited. 

Eo  =  120  volts,  d.c.;  Ei  =  5  volts;  7i  =  19.5  amps.;  R  =56.1  ohms;  G  =  0; 
L  =  0.418  henrys;  C  =  3.053  microfarads;  timing  wave  100  cycles. 

For  the  circuit  in  Fig.  94  a  resistance  equal  to  the  surge 
impedance  of  the  circuit,  VL/VC,  is  inserted  at  the 
receiver  end  of  the  line.  All  the  energy  of  the  traveling 
wave  was  dissipated  into  heat  by  the  Ri^  losses  at  the 
receiver  end  of  the  line  and  as  a  consequence  there  was  no 
reflected  voltage  or  current  waves  or  return  flow  of  power. 
From  the  timing  wave  and  known  length  of  line  it  is  found 
that  the  velocity  of  propagation  of  the  impulse  is  equal  to 


118 


ELECTRIC  TRANSIENTS 


v  or  3-1010  cm.  per  second,  the  velocity  of  propagation  of 
an  electromagnetic  field  in  free  space. 

A  traveling  wave  in  an  electric  line  is  sometimes  trans- 
formed into  a  standing  wave,  as  illustrated  by  the  oscillo- 
grams  in  Figs.  95,  96  and  97.  In  Fig.  95,  with  the 
receiver  end  of  the  line  open,  both  the  voltage  and  current 
waves  show  that  the  traveling  wave  passed  from  the  genera- 
tor to  the  receiver  end  of  the  line  and  back  again  four 
times  before  it  was  changed  into  a  standing  wave.  During 
this  period  the  voltage  and  current  waves  are  in  phase  or 


FIG.  93. — Traveling  waves  on  artificial  transmission  line.     Receiver  end  open. 
Eo  =  120  volts  d.c.;  Ei  =  5  volts;  /i  =  19.5  amps.  R  =56.1  ohms;  G  =  0; 
L  =  0.418  henrys;  C  =  3.053  microfarads;  timing  wave  100  cycles. 

180  deg.  apart,  showing  a  flow  of  power  along  the  line,  but 
when  the  traveling  wave  is  changed  to  an  oscillation  the 
current  leads  the  voltage  (note  position  of  vibrators  in 
the  circuit  diagram)  by  90  deg.  If  the  current  leads  or 
lags  90  deg.  with  respect  to  the  voltage,  the  power  in  the 
circuit  is  reactive  and  therefore  the  average  flow  of  power 
along  the  line  is  equal  to  zero. 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      119 

Similarly,  the  oscillograms  in  Figs.  96  and  97  show 
impulses  which  after  passing  over  the  lines  several  times 
as  traveling  waves  are  transformed  into  standing  waves  or 
oscillations.  In  each  case  the  impulse  starts  as  a  traveling 
wave  with  the  current  and  voltage  in  phase  and  a  flow  of 
power  along  the  line.  The  oscillogram  shows  that  the 
traveling  wave  was  converted  into  an  oscillation  or  stand- 
ing wave,  in  which  the  current  and  voltage  differ  by  90  deg. 
in  time  phase,  in  less  than  one  hundredth  of  a  second,  and 
that  the  energy  then  oscillated  between  the  magnetic  and 
dielectric  fields  without  flow  of  power  along  the  line. 


FIG.  94. — Traveling  waves  on  artificial  transmission  line. 

Receiver  resistance  =\/L /\/C;  Eo  =  120  volts  d.  c.;  Ei  =  5  volts;  I\  =  19.5 
amps.;  R  =  56.1  ohms;G=  0;  L  =  0.418  henrys;  C  =  3.053  microfarads;  timing 
wave  100  cycles. 

In  Fig.  97  the  vibrator  connections  for  the  voltage  wave, 
t>3,  were  reversed;  the  voltage  and  current  were  in  phase 
instead  of  180°  apart  as  indicated  by  the  oscillogram. 

The  change  in  frequency  when  the  traveling  wave  is 
converted  into  a  standing  wave  should  be  noted.  In  the 


120  ELECTRIC  TRANSIENTS 

traveling  wave  the  inductance  and  condensance  of  the 
line  alone  determines  the  velocity  of  propagation  while 
for  the  oscillations  or  standing  waves  the  line  and  trans- 
former oscillate  together  as  a  compound  circuit. 

In  determining  the  instantaneous  values  for  the  current 
and  voltage  at  any  point  on  the  system  the  power  flow  must 
be  taken  into  consideration  in  addition  to  the  dissipation 
of  the  transient  electric  energy  into  heat  as  expressed  by 
the  damping  factor  e~ut.  It  is  evident  that  the  flow  of 
power  may  be  increasing,  decreasing  or  unvarying  in  the 
direction  of  propagation. 

If  the  power  flow  is  uniform  the  expressions  for  the  cur- 
rent and  voltage  are  in  the  simplest  form  (244),  (255),  as 
the  power  transfer  factor  does  not  appear  in  the  equations. 

i  =  Io€~ut  cos  (ut  +  </>£  —  7)  (244) 

e  =  Eoe~ut  cos  (ut  +  <f>x  -  7)  (245) 

p  =  #0/oe-«<[l  --  sin2  (ut  +  4>x  -  7)]          (246) 

F1  J 
Average  power,  p  =  —~°  e~2ut  (247) 

Uniform  flow  of  transient  power  is  infrequent  but  may 
occur  in  special  cases.  Thus  if  a  transformer  line  and  load, 
as  in  Fig.  100,  are  disconnected  from  the  power  supply  and 
left  to  die  down  together,  uniform  flow  of  power  in  the  line 
may  be  realized  provided  the  dissipation  constant  of  the 
line  is  equal  to  the  average  dissipation  constant  of  the 
whole  system.  Consider  the  transformer  as  having  stored 
in  the  magnetic  field  a  comparatively  large  quantity  of 
energy  while  its  resistance  and  conductance  are  relatively 
small  compared  to  the  corresponding  value  for  the  line. 
Likewise  assume  that  the  load  part  of  the  circuit  has  very 
little  energy  stored  in  its  magnetic  and  dielectric  fields  and 
that  its  dissipation  constant  is  large  as  compared  to  that 
of  the  line.  Under  these  conditions  the  dissipation  of 
energy  is  most  rapid  in  the  load  part  of  the  circuit  and 
slowest  in  the  transformer.  Hence  a 'flow  of  energy  will 
occur  from  the  transformer  to  the  load.  If  the  rate  of 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      121   - 

energy  dissipation  of  the  line  is  midway  between  the  corre- 
sponding rates  for  the  load  and  transformers  the  energy 
dissipated  in  the  line  would  be  equal  to  the  amount  initi- 
ally stored  in  the  line  while  part  of  the  energy  originally 
stored  in  the  transformer  flows  through  the  line  and  is  dissi- 
pated in  the  load  part  of  the  circuit.  The  flow  of  power  in 
the  line  would  be  uniform  as  it  delivers  to  the  load  part  of 
the  circuit  all  the  energy  received  from  the  transformer. 


FIG.  95. — Traveling  waves  changing  to  standing  waves  on  artificial  transmission 

line. 

R  =  55.32  ohms;  G  =  0;  L  =  0.419  hemys;  C  =  3.05  microfarads;  Length  = 
207  miles;  4/0  copper;  96  in  spacing;  Transformer  L  =  37.8  henrys;  timing  wave 
100  cycles.  , 

The  flow  of  power  decreases  along  the  line  in  the  direction 
of  propagation,  if  energy  is  left  in  the  circuit  elements  as 
the  traveling  wave  passes  along  the  line.  That  is,  the 
traveling  wave  scatters  part  of  its  energy  along  its  path 
and  thus  decreases  in  intensity  with  the  distance  traveled. 
This  decrease  is  expressed  by  a  power  transfer  constant,  s, 
comparable  to  the  power  dissipation  constant  u.  If  no 
energy  were  supplied  to  the  line  by  the  traveling  wave  the 


122  ELECTRIC  TRANSIENTS 

voltage  and  current  would  decrease  by  the  dissipation 
factor  t~ut.  With  power  supplied  by  the  flow  of  energy 
the  decrease  would  be  slower  and  would  be  expressed 
by  a  combination  of  the  damping  and  power  transfer 
factors. 

For  decreasing  flow  of  power: 

Damping  factor  =  e~ut  (248) 

Power  transfer  factor  =  e+st  (249) 

Combined  damping  and  power  transfer  factor 

'  (250) 

Similarly  if  the  flow  of  power  increases  along  the  line  in 
the  direction  of  propagation  the  traveling  wave  receives 


FIG.  96. — Traveling  waves  changing  to  standing  waves  of  artificial  transmission 

line. 

Eo=  110  volts;  7i=  19.8  amps.;  R  =  52.9  ohms;  G  =  0;  L  =  0.412    henrys; 
C  =  3.03  microfarads;  timing  wave  60  cycles. 

energy  from  the  line  elements  and  the  actual  decrease  in 
the  voltage  and  current  is  greater  than  indicated  by  the 
dissipation  constant.  The  power  transfer  would  in  this 
case  be  negative,  and  the  combined  damping  and  power 
transfer  factor  would  be  expressed  by  equation  (253). 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      123 

For  increasing  flow  of  power : 

Damping  factor  =  e-ut  (251) 

Power  transfer  factor  =  e~st  (252) 

Combined   damping  and  power  transfer  factor 

=  e-(M  +  s)'     (253) 

To  express  the  instantaneous  values  of  the  current  and 
voltage  at  any  point  in  the  circuit  a  distance  factor  must 
be  included.  For  if  the  traveling  wave  either  scatters  or 
gathers  in  energy  as  it  travels  along  the  line  the  voltage  and 
current  factors  decrease  at  a  lesser  or  greater  rate,  as  the 
case  may  be,  in  the  direction  of  propagation  than  if  the  flow 
of  power  were  uniform.  In  order  to  use  only  one  power 
transfer  constant,  s,  in  the  equation,  let  X  =  the  distance  x 
expressed  in  velocity  measure  (254) 

For  decreasing  flow  of  power  along  the  line : 

the  distance  damping  factor  =  e~'  (255) 

For  increasing  flow  of  power  along  the  line : 

the  distance  damping  factor  =  esX  (256) 

The  instantaneous  values  of  the  transient  current,  voltage 
and  power  under  conditions  producing  a  flow  of  power  along 
the  line  from  the  point  of  reference,  in  the  direction  of  propa- 
gation may  be  expressed  by  equations  (257),  (258),  or 
(259),  (260). 

i  =  I0e~(      °'  e  +  ?X  cos  (cot  +  0  X  --  T)  (257) 

e  =  E0e          "c        cos  (cot  +  0X  —  7)  (258) 

i  =  Le~ut e  ±       °  cos  (cot  +  0X  -  7)  (259) 

e  =  E0e  ~  ute~       °  cos  (cot  +  </>X  --  7)  (260) 

T  -2ut       ±2s(t  -  X)  . 

p  =  loEoe         e  [I   —  Sin2  (cot  +    0X  —  7)! 

(261) 

Average  power,  p  =  Lfj>r**€±™-»  (262) 

2i 

The  upper  sign  of  4>X  applies  to  waves  traveling  in  the 
direction  of  increasing  values  of  X  and  the  lower  sign  for 
returning  waves,  for  which  X  is  decreasing.  For  s  =  0 


124 


ELECTRIC  TRANSIENTS 


which  represents  a  constant  flow  of  power,  equations 
(259)  and  (260)  become  identical  with  equations  (244)  and 
(245).  Referring  to  Fig.  100,  already  used  for  illustrating 
the  flow  of  constant  power,  it  is  evident  that  if  the  dissipa- 
tion constant  for  the  line  is  less  than  the  average  dissipa- 
tion constant  for  the  system  the  flow  of  power  from  the 
transformer  will  be  such  as  to  increase  the  power  stored 
in  the  line,  while  if  the  line  dissipation  constant  is  greater 
than  the  average  the  reverse  would  be  the  case. 


FIG.    97. — Traveling  waves  changing  to  standing  waves  on  artificial  transmission 

line. 
Eo  =  120  volts;  Length  =  200  miles;  4/0  copper;  120  in.  spacing;  timing  wave 

100  cycles. 

Traveling  waves  are  of  very  frequent  occurrence  in  elec- 
tric power  systems. .  Not  merely  such  violent  disturbances 
as  direct  strokes  of  lightning  or  short  circuits,  but  practi- 
cally every  change  in  load  or  circuit  conditions  produce 
transient  waves  that  travel  over  the  system.  Simple  travel- 
ing waves  as  illustrated  by  the  oscillograms  in  Figs.  92  to 

101  are  frequently  called  impulses.     In  the  first  part  of  the 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      125 


FIG.   98. — Oscillation  of    compound    circuit.     Starting    transient   of    artificial 

transmission  line  and  step-up  transformer. 

Length  of  line  =  52  miles;  4/0  copper;  96  in.  spacing,  R  =  13.84  ohms  ;  G  = 
0;  L  =  0.105  henrys;  C  =  0.764  microfarads;  transformer  L  =  37.8  henrys;  60 
cycle  supply. 


FIG.  99. — Oscillation    of    compound    circuit.     Starting   transient   of    (artificial 

transmssion)  line   and  transformers. 

Length  of  line  52  miles;  4/0  copper;  96  in.  spacing;  R   =  13.84  ohms;  G  =  0; 
L  =  0.105  henrys;  C  =  0.764  microfarads;  60  cycle  supply. 


126  ELECTRIC  TRANSIENTS 

impulse  as  it  passes  along  a  line  the  wave  energy  increases 
at  a  rate  depending  on  the  steepness  of  the  wave  front,  and 
after  the  maximum  value  is  reached  the  wave  energy 
decreases.  While  the  wave  energy  increases  the  combined 
dissipation  and  power  transfer  factor  is  represented  by 
c~'  '*  as  in  equation  (253),  and  during  the  decreasing 
stage  by  e  }t  as  in  equation  (250).  The  steepness  of 

the  wave  front  which  corresponds  to  the  sharpness  or 
suddenness  of  a  blow  is  often  a  more  important  factor  in 
causing  damage  to  the  electric  system  than  the  quantity 
of  energy  involved. 

Compound  Circuits. — In  commercial  systems  the  trans- 
mission line  is  not  an  independent  unit  but  merely  a  link 
between  the  generator  and  load  circuits.  Step-up  and 
step-down  transformers,  generators  and  load  circuits, 
lightning  arresters  and  regulating  devices,  and  all  the 
apparatus  necessary  for  the  operation  of  the  system  are 
electrically  interconnected  into  one  unit.  In  the  several 
parts  of  the  system  the  circuit  constants  differ  in  relative 

magnitude  and  hence  the  velocity  of 
propagation  of  an  electric  impulse 
varies  and  no  two  sections  may  have 
the  same  natural  period  of  oscilla- 
tion. While  the  whole  system  may 


FIG.  100.— circuit  diagram     oscillate   as   a   unit   partial   oscilla- 

of  a  compound  circuit.  , .  ,.  ,  ,. 

tions   are    of   much   more   frequent 
occurrence. 

In  Figs.  98,  99,  101  and  102  are  shown  the  oscillations  of 
compound  circuits  consisting  of  an  artificial  transmission  line 
and  transformers.  The  ripples  on  the  current  wave,  Vi,  indi- 
cate a  wave  traveling  over  the  transmission  line  alone.  From 
measurements  on  the  film,  Fig.  101,  the  length  of  the  line  is 
found  to  be  207  miles.  The  line  and  transformers  oscillate 
as  a  compound  circuit  at  a  frequency  of  10.5  cycles  per 
second.  In  Fig.  102  the  length  of  the  second  half  wave  is 
longer  than  for  the  first  half  wave.  This  is  due  to  a  varia- 
tion in  the  permeability  of  the  iron  in  the  transformer  core. 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      127 


§oi 

60 

I 


en 

sl 

§& 

O  iO 


B  " 


03  .0 


II 

fl     ft 

S  o 


128 


ELECTRIC  TRANSIENTS 


•^  g 

"• 


^3   O 


fl  >> 

g'a 
+3  a 


<Jo 
to 


3  ,y 

o   o 
*-i    03 

'3  a 


o  o 
fl" 

§a 
«§ 

^    o 

a 
o 


OSCILLATIONS,  SURGES  AND  TRAVELING  WAVES      129 

Problems  and  Experiments 

1.  Given  a  transmission  line,  80  miles  long,  of  No.  0000  copper,  spaced 
12  ft.  and  with  the  receiver  end  open.     From  handbook  tables  obtain  the  line 
constants.     Find  the  fundamental  oscillation  frequency  of  the  line.     Check 
the  results  by  solving  for  the  frequency  from  the  known  velocity  of  propaga- 
tion of  an  electric  wave  in  space  and  use  the  given  length  of  the  line. 

2.  Make  a  series  of  oscillograms  similar  to  Figs.  84,  86,  88  and  90,  on  an 
artificial  transmission  line.     From  the  oscillograms  determine  the  equiva- 
lent length  of  actual  line.     Check  by  determining  the  natural  frequency  of 
oscillation  from  the  line  constants. 

3.  From  the  oscillogram  in  Fig.  95  or  97  determine  the  frequency  of  oscil- 
lation of  the  transmission  line  alone  and  the  transmission  line  combined 
with  the  transformer.     Assume  the  condensance  of  the  transformer  equal  to 
zero.     From  the  data  given  calculate  the  inductance  of  the  transformer. 

4.  In  the  oscillogram  in  Fig.  101  the  ripples  on  the  voltage  wave  indicate 
reflections  of  traveling  waves  in  the  transmission  line  with  the  receiver  end 
open.     Calculate  the  length  of  the  line. 

6.  From  the  oscillograms  and  data  in  Fig.  101  calculate  the  inductance 
in  the  transformer  in  the  compound  circuit.  Assume  the  condensance  of 
the  transformer  equal  to  zero.  It  should  be  noted  that  the  inductance  is 
essentially  massed  while  the  condensance  is  distributed  and  hence  for  the 

combined  circuit/  =        /   - 

^  v    2irLC 

6.  From  the  data  given  in  Fig.  102  calculate  the  average  inductance  of 
the  transformers  during  the  first  half  cycle  after  the  current  and  voltage 
wave  lines  cross;  also  during  the  second  half  cycle. 

7.  Make  oscillograms  of  the  oscillations  of  compound  circuits,  similar  to 
Figs.  99,  100,  101,  and  102. 


CHAPTER  VII 
VARIABLE  CIRCUIT  CONSTANTS 

In  the  preceding  chapters  the  fundamental  laws  of  tran- 
sient electric  phenomena  are  derived  under  the  assumption 
that  in  any  given  circuit  the  resistance,  inductance,  con- 
ductance and  condensance,  the  so-called  circuit  constants, 
remain  constant  in  value  during  the  transition  period  under 
discussion.  The  transients  are  due  to  changes  in  circuit 
condition  or  in  the  impressed  voltage,  but  during  the  period 
required  for  the  dissipation  of  the  stored  energy,  or  the 
readjustment  of  the  energy  content  in  the  system  the  values 
of  R,  L,  G  and  C  are  assumed  constant.  The  oscillograms, 
Chaps.  Ill  to  VI  inclusive,  of  electric  transients  were 
obtained  from  circuits  in  which  the  resistance,  inductance, 
conductance  and  condensance  remained  essentially  constant. 

It  is  evident  that  if  the  circuit  constants  do  not  remain 
constant  during  the  period  the  transients  occur  but  vary 
rapidly  over  a  wide  range  of  values  the  nature  of  the  result- 
ing electric  phenomena  must  be  correspondingly  more 
complex.  The  laws  for  the  variations  in  R,  L,  G  and  C  are 
not  always  known  or  are  so  complex  that  they  can  not  be 
represented  in  the  form  of  equations.  For  example,  data 
for  the  quantitative  ratios  between  the  magnetomotive 
force  and  the  resulting  magnetic  flux  in  iron  clad  circuits, 
as  indicated  by  the  hysterises  loop,  may  readily  be  obtained 
experimentally  but  it  has  not  been  possible  to  express  the 
relation  in  the  form  of  a  mathematical  equation.  The 
empirical  equations  in  common  use  are  limited  in  their 
application  and  give  only  approximate  values. 

Variable  Resistance. — Change  in  temperature  is  the 
most  important  factor  in  producing  variations  in  the  resis- 
tance of  electrical  conductors,  the  R  circuit  constant.  For 

130 


VARIABLE  CIRCUIT  CONSTANTS 


131 


metals  the  specific  resistance  is  a  linear  function  of  the 
temperature  over  a  fairly  wide  range. 

----  po  +  at0  (270) 

=  specific  resistance  at  t°  C. 
—  specific  resistance  at  0°  C. 
=  temperature  coefficient. 


Pt 


PO 


For  rapid  changes  in  temperature  the  rate  of  change  in 
the  resistance  may  be  large.  This  is  illustrated  by  the 
oscillograms  in  Figs.  103,  104.  For  the  tungsten  incandes- 
cent lamp,  Fig.  103,  a  starting  transient  appears  in  the 
current  due  to  a  rapid  increase  in  the  resistance  of  the  fila- 


FIG.   103. — Starting  current  transient  of  a  60-watt,  120  volts,  tungsten  incan- 
descent lamp.     Resistance  variable;  timing  wave  100  cycles. 

ment  as  the  temperature  rises.  When  the  switch  is  closed 
the  filament  is  at  room  temperature  and  the  resistance  low. 
The  current  flowing  through  the  lamp  rapidly  heats  the 
filament  to  incandescence  with  an  accompanying  increase  in 
the  resistance  and  a  decrease  in  the  current.  The  timing 


132 


ELECTRIC  TRANSIENTS 


wave  shows  that  it  required  about  0.02  of  a  second  for  the 
lamp  to  reach  full  brilliancy.  During  this  period  the  resis- 
tance of  the  filament  increased  by  400  per  cent  of  its  initial 
value. 

For  carbon  the  resistance  decreases  with  an  increase  in 
temperature,  or  the  temperature  coefficient  is  negative,  as 
illustrated  in  Fig.  104,  showing  that  the  time  required  for 
the  resistance  to  reach  a  constant  value  was  approximately 
0.5  of  a  second  and  that  the  resistance  of  the  incandescent 
carbon  filament  is  about  70  per  cent  of  its  value  at  room 
temperature. 


FIG.   104. — Starting  current  transient  of  a  50  watt,  120  volts,  carbon  incandescent 
lamp.     Resistance  variable;  timing  wave  100  cycles. 

The  temperature  of  the  lamp  filament  increases  until  the 
dissipation  of  heat  by  radiation  from  the  lamp  is  equal  to 
the  heat  generated  by  Ri2  losses.  For  a  direct  current  sup- 
ply with  constant  impressed  voltage  the  constant  tem- 
perature condition  is  quickly  reached.  For  alternating 
currents  the  power  supplied  to  the  lamps  pulsates  with  double 


VARIABLE  CIRCUIT  CONSTANTS  133 

the  current  frequency  and  as  the  lamp  emits  or  radiates 
heat  continuously  the  temperature,  and  therefore  the  resis- 
tance of  the  filament,  pulsates.  This  is  illustrated  by  the 
oscillogram  in  Fig.  105.  Alternating  currents  are  impressed 
on  two  pairs  of  tungsten  and  carbon  lamps,  arranged  as 
shown  in  the  circuit  diagram,  with  the  vibrator  of  the  oscil- 
lograph in  the  bridge  connection.  Since  the  resistance  of 


FIG.    105. — Pulsating    resistance    of    tungsten    and  carbon  lamps,   alternating 
currents;  60  cycle  supply. 

the  tungsten  lamp  increases  and  the  carbon  lamp  decreases 
with  an  increase  in  temperature,  the  pulsations  in  the  Ri2 
losses  unbalance  the  bridge  as  indicated  by  the  pulsations 
in  the  currents  flowing  through  the  vibrator. 

The  resistance  of  the  electric  arc  depends  on  many  factors 
and  may  vary  over  a  wide  range  with  extreme  rapidity. 
Since  the  resistance  of  the  arc  decreases  with  the  increase 
in  temperature  the  arc  alone  is  unstable  and  hence  must  be 
provided  with  a  " ballast"  to  make  continuous  operation 
possible.  On  alternating  currents  an  inductance  placed  in 


134 


ELECTRIC  TRANSIENTS 


series  with  the  arc  serves  as  the  stabilizer  and  the  variations 
in  the  resistance  of  the  arc  are  counterbalanced  by  the 
induced  voltage  in  the  inductance.  In  direct  current  arc 
lamps  a  series  resistance  serves  the  same  purpose. 

In  commercial  systems  the  electric  arcs  that  affect  the 
series  resistance,  the  R  circuit  constant,  occur  chiefly  in 
the  opening  of  switches.  In  breaking  the  circuit  under 
load,  especially  when  a  large  quantity  of  energy  is  stored 
magnetically  in  the  circuit,  arcs  form  in  which  the  resistance 
varies  rapidly  from  zero  at  start  to  infinity  when  the  circuit 
is  open. 


^^^^^^^^^^•^MM^^^^MMM^^^^^^MMM^M^^^M 

FIG.   106. — Transformer  magnetizing  current;  no  starting  transient. 
Vi   =  106  volts;  v-2  =  primary  current;  03,  calibration  current  =  10.0  amps.;  10 
KVA.  transformers;/  =  60  cycles. 

This  is  illustrated  by  the  oscillogram  in  Fig.  110.  In 
the  opening  of  the  switch  an  arc  forms  whose  resistance 
rapidly  increases,  approaching  infinity  when  the  circuit 
opens,  which  occurs  at  the  point  of  maximum  value  in  the 
voltage  curve.  The  increase  in  the  resistance  can  be  deter- 
mined quantitatively  from  the  oscillogram  by  combining 


VARIABLE  CIRCUIT  CONSTANTS  135 

data  from  the  rapidly  increasing  voltage  and  decreasing 
current  curves. 

Variable  Inductance. — In  iron-clad  circuits  as  in  trans- 
formers the  magnetic  flux  is  not  directly  proportional  to  the 
ampere  turns  or  magnetizing  force.  Hence  the  inductance, 
the  L  circuit  constant  is  not  constant  but  varies  with  the 
permeability  of  the  iron.  Moreover,  the  variation  in  the 
inductance  is  different  for  decreasing  and  increasing  flux 


FIG.   107. — Starting  transient  of  magnetizing  current  in  iron-clad  circuit. 
vi,    primary  voltage  =  236  volts;  vz  =  primary  current;  %  calibration  current 
=  13.0  amps.;  10  KVA..  transformers;/  =  60  cycles. 

values  and  depends  on  the  maximum  flux  density  as  indi- 
cated by  the  form  of  the  hysteresis  loop.  As  no  satisfac- 
tory mathematical  expression  has  yet  been  found  for  the 
hysteresis  cycle,  solutions  of  practical  problems  are  obtained 
by  a  series  of  approximations.  As  a  first  step  in  obtaining 
the  shape  of  transients  in  iron-clad  circuits,  neglecting  the 
difference  between  increasing  and  decreasing  flux  values, 
Frohlich's  formula  is  generally  used. 

HB  =  v  +  *H  (271) 


136 


ELECTRIC  TRANSIENTS 


The  formula  is  based  on  the  assumption  that  the  permea- 
bility of  the  iron  is  proportional  to  its  remaining  magnetiza- 
bility  and  states  that  the  reluctivity  of  an  iron-clad  circuit 
is  a  linear  function  of  the  field  intensity. 


FIG.   108. — Starting  transient  of  magnetizing  current  in  iron-clad  circuit. 
»i,    primary  voltage  =  106  volts;  02,  primary  current;  03,  calibration  current 
=  10.0  amps.;  10  KVA.  transformer;  /  =  60  cycles. 

The  effect  of  variable  inductance  in  iron-clad  circuits 
may  be  illustrated  by  the  starting  transients  of  alternating 
current  transformers.  The  magnitude  of  the  starting 
current  transient  depends  more  on  conditions  affecting  the 
value  of  the  inductance  in  the  circuit  than  on  what  point 
on  the  voltage  cycle  the  switch  is  closed.  The  direction 
and  magnitude  of  the  residual  magnetism  are  important 
factors  as  a  combination  of  much  residual  flux  with  an 
additional  magnetizing  force  in  the  same  direction  may 
bring  the  flux  density  in  the  core  beyond  the  saturation 
point  and  hence  greatly  reduce  the  inductance  in  the 
circuit. 

For  the  oscillograms  in  Figs.  106  to  109  a  constant  alter- 
nating current  voltage  of  sine  wave  shape  was  impressed  on 


VARIABLE  CIRCUIT  CONSTANTS  137 

the  transformer  terminals.  In  Figs.  106,  107  and  108  the 
residual  magnetism  in  the  iron  core  was,  in  each  case, 
removed  before  the  oscillogram  was  taken.  The  three 
oscillograms  form  a  series  showing  the  transient  current  due 
to  the  closing  of  the  switch  at  different  points  of  the  voltage 
cycle.  In  Fig.  106  the  switch  was  closed  at  an  instant  the 
magnetizing  current  would  have  been  zero  (maximum  point 
on  the  voltage  wave),  if  the  circuit  had  been  closed  earlier, 


FIG.   109. — Starting   transient   in   transformer   magnetizing    current.     Residual 

magnetism. 

vi,  primary  voltage  =  150  volts;  02,  primary  current;  vs,  calibration  current 
=  2.5  amps.;  /  =  60  cycles. 

and  hence  no  starting  transient.  In  Figs.  107  and  108  the 
switch  was  thrown  at  other  than  the  zero  point  of  the  mag- 
netizing current  cycle.  The  impressed  voltage  was  less  than 
normal  and  the  change  in  the  flux  density  is  not  large  and 
hence  the  inductance  at  the  maximum  points  of  the  mag- 
netizing current  wave  is  practically  constant.  The  starting 
current  transients  under  the  given  conditions  may  be  expres- 
sed by  an  exponential  equation  as  explained  in  Chap.  IV. 


138 


ELECTRIC  TRANSIENTS 


The  starting  transient  in  Fig.  109  differs  greatly  both  in 
form  and  magnitude,  as  compared  to  Fig.  108,  although  the 
circuits  were  closed  in  the  two  cases  at  approximately  the 
same  point  on  the  voltage  wave.  In  Fig.  109  the  impressed 
voltage  was  higher  than  the  rating  of  the  transformer  and 
the  residual  magnetism  in  the  iron  core  was  in  the  same 
direction  as  the  flux  produced  by  the  magnetizing  current 
during  the  first  half  cycle.  Above  saturation  of  the  iron 


FIG.   110.— Breaking     generator     field     circuit.     Field     current     and     voltage 

transients. 

vi  =  100  cycle  timing  wave;  v?,  impressed  voltage  =31.5  volts;  va,  field  cur- 
rent =  4.0  amps. 

core  the  transformer  inductance  is  relatively  small  and 
hence  the  first  half  cycle  shows  a  correspondingly  large 
current  transient.  A  smooth  curve  drawn  through  the 
successive  maximum  values  of  the  starting  transient  in 
Figs.  107  or  108  could  with  a  fair  degree  of  accuracy  be 
expressed  by  the  exponential  equation ;  but  the  correspond- 
ing curve  drawn  through  the  successive  maximum  values 
of  the  current  wave  in  Fig.  109  would  have  a  much  steeper 


VARIABLE  CIRCUIT  CONSTANTS 


139 


gradient  due  to  the  variation  in  the  inductance,  L,  of  the 
transformer  winding. 

The  same  effect,  due  to  variable  inductance,  may  be 
obtained  in  breaking  the  field  circuit  of  a  direct  current 
generator  as  illustrated  by  the  oscillogram  in  Fig.  110. 
The  change  in  the  voltage  and  current  curves  from  the 
instant  the  jaws  of  the  switch  separate  to  the  peak  value 
of  the  voltage  is  largely  due  to  a  change  in  the  arc  resistance. 
After  the  arc  breaks,  at  the  peak  of  the  voltage  curve,  the 


FIG.   111. — Building  up  generator  field.     Field   current  and  armature  voltage 

transients. 

vi  =  generator  terminal  voltage;  vz  =  field  current;  vs  —  100    cycle    timing 
wave. 

vibrator  circuit  provides  a  path  for  the  dissipation  of  the 
energy  stored  in  the  field.  As  the  resistance  in  the  vibrator 
circuit  is  constant  the  voltage  curve  also  represents  the 
transient  current.  The  dotted  curve  traced  on  the  oscillo- 
gram shows  the  exponential  curve  conforming  with  the 
latter  part  of  the  actual  voltage  or  current  curves.  The 
relative  magnitude  of  the  peak  value  of  the  voltage  to  the 


140 


ELECTRIC  TRANSIENTS 


corresponding  initial  value  of  the  dotted  curve  indicates 
the  change  in  magnitude  of  the  inductance  in  the  field 
winding. 

The  corresponding  variation  in  the  inductance  when  the 
generator  field  is  formed  is  evidenced  by  the  starting  field 
current  and  armature  voltage  curves  shown  in  Fig.  111. 


FIG.  112. — Arcing  grounds  on  transmission  line. 

Ground  at  generator  end.     Impressed  voltage  =  90  volts;/  =  60  cycles;  ^2  = 
arc  voltage;  03  =  arc  current. 

Variable  Conductance. — In  the  calculations  on  power 
transmission  lines  and  in  general  for  constant  potential 
systems  in  good  condition  the  leakage  through  the  insulation 
is  small,  so  that  the  conductance  is  negligible  and  the  G 
circuit  constant  may  be  taken  as  equal  to  zero.  The  insula- 
tion of  electric  circuits  deteriorate  with  varying  rates  and 
the  conductance  and  leakage  increase  and  may  become  very 
large,  as  for  example,  if  the  insulation  completely  breaks 
down  and  a  short  circuit  is  formed.  A  rupture  of  the  insula- 
tion or  any  sudden  change  in  the  conductance  of  the  electric 
circuit  will  of  necessity  cause  violent  disturbances  in  the 


VARIABLE  CIRCUIT  CONSTANTS 


141 


FIG.   113. — Arcing- grounds  on  transmission  line. 

Semi-continuous  copper-carbon  arc  114  miles  from  generator  end  of  207 
mile  artificial  transmission  line.  4/0  copper,  96  in.  spacing.  v\  =  arc  voltage; 
#2  =  arc  current;  vs  —  line  current. 


FIG.   114. — Arcing  grounds  on  transmission  line. 

Arc  at  receiver  end.     vi  =  100  cycles  timing  wave;  vz  =  current  receiver  end: 
=  voltage  receiver  end. 


142 


ELECTRIC  TRANSIENTS 


system.  Arcing  grounds  or  intermittent  arcs,  as  illustrated 
by  the  oscillograms  in  Figs.  112  to  115,  are  prolific  sources 
of  electric  transients.  It  is  evident  that  momentary 
short  circuits,  as  would  be  produced  by  an  intermittent 
arcing  ground  with  the  conductance  varying  practically 
from  zero  to  infinity  at  an  extremely  rapid  rate,  would  give 
rise  to  oscillations  of  any  frequency  and  produce  waves  and 
impulses  that  would  travel  to  all  parts  of  the  system. 


FIG.   115. — Arcing  grounds  on  transmission  line. 

Arc  at  middle  of  line.  v\  —  100  cycle  timing  wave;  vi  =  arc  current;  va  —  arc 
voltage. 

Variable  Condensance. — Under  ordinary  conditions  and 
for  low  voltages,  air  is  very  nearly  a  perfect  insulator.  In 
other  words,  the  conductivity  of  air  is  practically  zero,  the 
permittivity,  unity  and  the  energy  loss  extremely  small. 
If  the  voltage  is  increased  until  the  limit  of  the  insulating 
strength  of  the  air  is  reached  important  changes  occur 
in  both  the  electric  and  dielectric  circuit  constants.  With 
the  occurrence  of  visual  corona  in  high  voltage  circuits  the 
conductivity  of  the  air  in  the  space  filled  by  the  corona  is 


VARIABLE  CIRCUIT  CONSTANTS 


143 


increased.  Thus  in  circuits  with  parallel  wires  as  high 
tension  transmission  lines  a  voltage  gradient  above  29.8  ky. 
per  cm.  will  produce  corona  in  the  air  surrounding  the 
conductor  surface  and  this  space  filled  by  the  corona  glow 
becomes  semi-conducting.  This  produces  a  change  in  the 
circuit  condensance  as  with  the  appearance  of  the  corona  the 
effective  size  of  the  conductor,  and  hence  of  the  condenser 
surface,  is  increased.  For  alternating  currents  the  visual 


FIG.   116. — Variable  condensance.     Corona. 

Single  phase  line  135  ft.  long,  10  in.  spacing,  No.  24  A.W.G.  steel  wire.      Line 
voltage  =  3400  volts;  line  current  =  0.0008  amps. 

corona,  is  intermittent,  appearing  only  near  the  peaks  of 
the  successive  voltage  waves,  when  the  instantaneous  volt- 
age gradient  exceeds  29.8  kv.  per  cm.,  the  required  value 
for  producing  visual  corona.  As  a  consequence  the  con- 
densance of  the  alternating  current  circuit  when  corona 
occurs  is  variable,  pulsating  with  double  the  frequency  of 
the  voltage.  This  is  illustrated  by  the  oscillograms  in 
Figs.  116,  117.  If  an  alternating  current  voltage  of  sine 
wave  shape  is  impressed  on  a  circuit  having  constant  con- 


144 


ELECTRIC  TRANSIENTS 


densance  the  charging  current  would  also  follow  the 
sine  law.  If  the  condensance,  the  C  circuit  constant,  varies 
during  the  voltage  cycle,  a  corresponding  change  is  produced 
in  the  wave  shape  of  the  charging  current. 


FIG.   117. — Variable  condensance.      Corona. 

Line  covered  with  snow  and  swaying  in  the  wind.     Line  constants  same  as 
for  Fig.  116. 


Problems  and  Experiments 

1.  Take  oscillograms,  similar  to   Figs.    106,    108  and   109,  showing  the 
starting  transients  of  transformers. 

2.  Take  oscillograms   showing   the   variable  condensance  of   an   arcing 
ground  for  direct  and  alternating  currents  on  a  transmission  line. 

3.  Take  oscillograms  similar  to  Figs.  116  and  117,  showing  the  change  in 
condensance  produced  by  corona. 

4.  Take  an  oscillogram  similar  to  Fig.  110,  showing  the  voltage  across 
the  terminals.     Compare  the  operating  voltage  with  the  maximum  value 
when  the  switch  is  opened. 


CHAPTER  VIII 

RESONANCE 

Electric  resonance  phenomena  have  essentially  perman- 
ent or  stable  characteristics  but  are  closely  related  to,  and 
frequently  accompanied  by,  true  electric  transients.  The 
conditions  required  for  producing  resonance  and  expres- 
sions for  the  frequency  at  which  resonance  occurs,  in  simple 
electric  circuits,  are  referred  to  in  Chap.  IV  in  connection 
with  the  derivation  of  the  equations  for  the  natural  fre- 
quency of  free  oscillations.  Resonance  in  an  electric  circuit 
implies  a  forced  oscillation  of  energy  between  the  magnetic 
and  dielectric  fields,  during  which  the  energy  dissipated  as 
heat  by  the  Ri2  and  Ge'2  losses,  is  supplied  from  some  outside 
source.  Distinction  is  usually  made  between  voltage  reson- 
ance occurring  in  series  circuits,  and  current  resonance  that 
may  be  produced  in  parallel  circuits. 

Voltage  Resonance.  —  In  series  circuits  voltage  resonance 
occurs  at  that  frequency  of  the  impressed  voltage  for  which 
the  impedance  of  the  circuit  is  a  minimum.  In  series  circuits, 
as  in  Fig.  118,  the  impedance  is  a  minimum  when  the  con- 
densive  and  inductive  reactances  are  equal. 

,x  =  cx;  27T/L  =  (275) 


/=        \  (276) 

2irVLC 


z  ----  VR2  +  (Lx  -  cx)2  =R  (277) 

Frequently  the  assumption  is  made  that  a  circuit  is  in 
resonance  when  the  current  and  the  impressed  voltage  are 
in  phase,  as  illustrated  by  the  vector  diagram  in  Fig.  119. 
For  straight  series  circuits  the  conditions  required  for  unity 
power  factor  of  the  power  supplied  to  the  circuit  are  iden- 
10  145 


146 


ELECTRIC  TRANSIENTS 


tical  with  the  requirements  for  minimum  impedance,  but 
in  complex  circuits  or  for  current  resonance  in  parallel 
circuits  this  is  not  always  the  case. 


FIG.   1 18. — Series  circuit  for  voltage  resonance. 

Equation  (276)  gives  the  optimum  condition  for  reson- 
ance in  series  circuits  for  given  values  of  the  R,  L  and  C,  the 
circuit  constants.  Resonance  phenomena  are,  however, 

Y 


FIG.   119. — Vector   diagram   for   voltage   resonance   in   series   circuit    Fig.    118. 

not  limited  to  the  exact  frequency  determined  by  equation 
(276),  but  persist  over  a  range  of  frequencies,  more  or  less 
sharply  defined,  depending  on  the  relative  magnitude  of 
the  resistance  and  the  inductive  or  condensive  reactance. 
The  voltage-frequency  relation  for  given  constant  values 
of  R,  L  and  C,  is  shown  in  Fig.  120.  The  feature  of  special 


RESONANCE 


147 


interest  is  the  large  increase  in  LE  and  CE,  the  voltages 
across  the  inductance  and  the  condensance  under  resonance 
conditions.  If  the  resistance  is  small  ,E  arid  CE  may  rise 


FIG.   120. — Voltage  resonance  for  series  circuit  as  in  Fig.  118 

to  many  times  the  value  of  the  impressed  voltage  En. 
Voltage  resonance  in  power  circuits  is  undesirable  as  the 
increase  in  voltage  above  the  normal  operating  value 
endangers  the  insulation. 

The  effect  of  varying  the  resistance  on  the  sharpness  of 
resonance  is  illustrated  by  Fig.  121.  The  smaller  the  resis- 
tance the  higher  and  sharper  the  voltage  and  current  reson- 
ance peaks.  The  sharpness  of  resonance  may  be  defined 
as  the  ratio  of  the  inductive  reactance  or  the  condensive 
reactance  at  resonance  frequency  to  the  resistance  in  the 
circuit. 

Sharpness  of  resonance  =  ^  =  CD  (278) 

ti       H 

Reactance  Curves. — Curves  in  rectangular  coordinates 
showing  graphically  the  changes  in  magnitude  of  the 


148 


ELECTRIC  TRANSIENTS 


inductive  reactance  and  the  condensive  reactance  produced 
by  varying  the  frequency  of  the  impressed  voltage  are  of 
much  value  for  giving  a  clear  insight  into  resonance  phe- 
nomena. The  ordinates  of  the  curves  in  Fig.  122  represent 
respectively  the  inductive  reactance,  Lx,  the  condensive 
reactance,  cx,  and  the  total  reactance,  x,  with  the  frequency 
of  impressed  voltage  as  the  other  variable  represented 


/    \ 


FIG.   121. — Resonance  curves  for  series  circuit  with  different  resistances. 

along  the  X  axis.  Since  resonance  occurs  when  the 
impedance  of  the  series  circuit  is  a  minimum,  the  resonance 
frequency  is  indicated  by  the  intersection  of  the  total 
reactance  curve,  in  Fig.  122,  with  the  X  axis. 

Current  Resonance. — Forced  oscillatory  transfer  of 
energy  between  dielectric  and  magnetic  fields  is  the  basis  of 
resonance  phenomena  in  parallel  circuits  in  much  the 
same  manner  as  in  series  circuits,  but  the  resultant  voltage 
and  current  values  are  different.  In  simple  parallel 
circuits,  as  illustrated  by  Figs.  123  and  127,  current  reson- 
ance occurs  at  that  frequency  of  the  impressed  voltage  for 
which  the  total  admittance  is  a  minimum.  In  discussions  of 
resonance  phenomena  it  is  frequently  assumed  that  the 


RESONANCE 


149 


conditions  for  current  resonance  in  parallel  circuits  are 
met  when  the  inductive  and  condensive  susceptances 
are  equal,  that  is,  when  the  impressed  current  and  voltage 
are  in  phase.  That  this  assumption  is  not  in  full  accord 
with  the  above  definition  of  current  resonance  for  all 


FIG.   122. — Reactance  curves.     Series  circuit. 

values  of  R  in  the  circuits  shown  in  Figs.  123  and  127,  may 
readily  be  seen  from  the  corresponding  vector  diagrams  in 
Figs.  124  and  128.  For  the  circuit  in  Fig.  123  current 
resonance  occurs  when  ,b  =  (b  under  the  condition  that 
R  =  0.  From  the  vector  diagram  in  Fig.  124: 


R  — 


J  ==  EQ(g  -  jjb)  ----  E0  ^ 

J  =  jcbEQ  =  juCE0 
I  ==  J  +  J  ==  E0[g  +j(cb  -  Lb)] 
I  =  1 


(279) 

(280) 
(281) 
(282) 


150 


ELECTRIC  TRANSIENTS 


The  total  current,  7,  will  be  in  phase  with  the  impressed 
voltage,  E,  if 

*  =  *•><*<>  =***  (284) 


Hence  for  unity  power  factor  supply,  the  frequency  for 
the  circuit  in  Fig.  123, 


~ 


(285) 


' 


FIG.   123. — Parallel  circuit  for  current  resonance. 


FIG.  124. — Vector  diagram  for  circuit  in  Fig.  123. 

For  maximum  current  resonance  the  total  admittance 
of  the  circuit  must  be  a  minimum  and  hence  for  constant 
impressed  voltage,  EQ,  the  total  current  must  be  a  minimum. 
Therefore,  the  resonance  frequency  may  be  obtained  by 


RESONANCE 


151 


equating  the  first  derivative  of  /  to  co,  L,  or  C,  as  the  case 
may  be,  in  equation  (283)  to  zero.  Taking  co  as  the  variable 
factor  with  R,  L,  C,  and  E  constants  for  the  circuit  in  Fig. 
123: 


Letting  C  be  the  variable  factor  with  R,  L,   co,  and  E 
constant: 

1      /I       "722 


/  = 


(287) 


FIG.   125. — Current  resonance.     Variable  u>.     For  Fig.  123,  Equation  (286). 

Letting  L  be  variable  with  R,  C,  co,  and  E  constants: 

(288) 


f  =  l  -1- 

J         o     \   or  r< 


UY 

V    C     / 


In  a  similar  manner  expressions  may  be  obtained  for 
unity  power  factor  frequency  and  maximum  current  resonance 
frequency  for  co,  C  or  L  respectively  as  the  variable  with  the 
other  factor  constants  for  the  circuit  in  Fig.  127. 


152 


ELECTRIC  TRANSIENTS 


j  =  E0(g  ~  jjb) 
J  ==  E0(G  +  job) 
t  --=  J+  J  ----  EQ[(g  +  G)  +  j(cb  -  Lb)} 


(289) 
(290) 
(291) 


FIG.   126. — Vector  diagram.     Variable  C.     For  Fig.    123,   equation  (287). 

The  total  current,  /,  will  be  in  phase  with  the  impressed 
voltage,  EQ  if 


E>9        |  97"   2 

it     -J-  CO  JLJ 


(293) 


Hence,  the  frequency  required  to  give  unity  powerfactor 
for  the  circuit  in  Fig.  127  is  the  same  as  for  Fig.  123. 


L2 


(294) 


The  frequency  for  maximum  current  resonance  if  w  is 
variable  while  R,  L,  (7,  G  and  E'o  are  constant,  Figs.  127,  128: 


RESONANCE  153 

If  C  be  the  variable,  while  R,  L,  G,  u  and  EQ  are  constant: 

(296) 


1     /  1  ~     R'2 


L'2 
If  L  be  the  variable,  while  R,  C,  G,  co  and  E^  are  constant: 

=  27r\      2LC      +  [r4  +        CL3  4L2C2 

(297) 


o 

o 

L° 

G          O 


FIG.   127.- — -Parallel  circuit  with  leaky  condenser. 

In  tuning  ratio  receiver  sets  resonance  is  obtained  by 
varying    C   or  L  as  expressed  by  equations   (296)    (297). 


FIG.   128. — Vector  diagram  for  circuit  in  Fig.  127. 

Changes  in  the  inductance  by  varying  the  number  of  turns, 
also    changes    the    ohmic   resistance   but   the    conditions 


154 


ELECTRIC  TRANSIENTS 


required  for  equation  (297)  may  be  obtained  experiment- 
ally for  circuits  in  which  the  change  in  L  may  be  produced 
by  varying  the  mutual  or  self-induction  between  parts  of 
the  inductance  in  circuit. 

The  smaller  the  resistance  in  the  resonating  circuit  the 
greater  the  increase  in  the  resonance  current  and  voltage. 
Resonance  phenomena  are  of  commercial  importance  only 
when  'the  resistance  in  circuit  is  small  as  compared  to  the 
inductance  and  condensance. 


FIG.   129. — Susceptance  curves  for  parallel  circuit. 

In  most  cases  and  particularly  those  of  greatest  impor- 
tance, the  resistance  is  negligibly  small.  If  R  and  G  are 
taken  equal  to  zero  all  the  resonance  frequency  equations 
(295)  to  (297)  become  identical  in  form. 

Resonance  frequency,  massed  circuit  constants  (approxi- 
mate value) : 


RESONANCE  155 

>  =  2.VLC  (298) 

In  commercial  work  equation  (298)  is  in  general  use,  giv- 
ing with  sufficient  accuracy  the  resonance  frequency  for 
simple  circuits  having  massed  condensance,  inductance  and 
resistance. 

For  distributed  circuit  constants,  as  in  long  transmission 
lines,  the  space  distribution  of  the  voltage  and  current 
waves  must  be  taken  into  consideration,  the  approximate 
resonance  frequency  is  given  by  equation  (299),  as  explained 
in  Chap.  VI  on  Transmission  Line  Oscillations. 

Resonance  frequency,  uniformly  distributed  circuit  con- 
stants (approximate  value) 

f  -  4VLC  (299) 

In  power  circuits  resonance  conditions  must  be  avoided 
or  the  resistance  in  circuit  be  sufficiently  large  to  prevent 
any  marked  increase  due  to  resonance  in  the  current  and 
voltage. 

Coupled  Circuits. — Resonance  phenomena  are  of  funda- 
mental importance  in  the  operation  of  radio  communica- 
tion apparatus.  The  circuits  in  commercial  use  are  more 
complex  than  the  forms  discussed  above  but  may  be  con- 
sidered as  combinations  of  simple  circuits.  In  general 
the  component  simple  circuits  have  certain  parts  in 
common. 

The  couplings  or  connections  may  be  made  in  a  number  of 
ways.  For  two  circuit  apparatus  the  coupling  is  generally 
made  in  one  of  the  following  ways: 

1.  By    direct    connection    across    an    inductance    coil. 
Direct  coupling  as  in  Fig.  130. 

2.  By  magnetic  induction.     Inductive  or  magnetic  coup- 
ling as  in  Fig.  131. 

3.  By    dielectric    induction.     Condensive,    capacitative 
or  dielectric  coupling  as  in  Fig.  132. 


loG 


ELEC TRIG  TEA NSIEN TS 


The  inductive  interaction  of  the  voltages  and  currents 
in  tAvo  resonating  coupled  circuits  and  the  transfer  of  the 


PTXRP — 1 — nRHT^-lf— 


M 


FIG.   130. — Direct  coupling. 

oscillating   energy   between    the    primary    and    secondary 
circuits  are  illustrated  by  the  oscillograms  in  Figs.  133  to 


FIG.   131.—  Inductive  or  magnetic  coupling. 


138.     The  oscillations  of  the  energy  between  the  dielectric 
and  magnetic  fields  of  each  circuit  are  combined  with  a 


FIG.   132. — Condensive  or  dielectric  coupling. 

rapid  to  and  fro  transfer  of  the  energy  between  the  mag- 
netically or  dielectrically  coupled  circuits.  In  Fig.  133 
the  energy  was  initially  stored  in  the  condenser  in  the  pri- 


RESONANCE  157 

mary  circuit.  By  closing  the  switch  oscillations  are  set  up 
between  the  dielectric  and  magnetic  fields  in  both  the 
primary  and  secondary  circuits,  and  these  are  combined 
with  a  rapid  to  and  fro  transfer  of  the  energy  between  the 
two  circuits.  The  oscillogram  shows  that  the  frequency 
of  oscillation  between  the  magnetic  and  dielectric  fields  in 
both  the  primary  and  secondary  was  790  cycles  per  second, 
while  the  frequency  of  transfer  between  the  circuits  was 
approximately  99  cycles  per  second.  That  is,  the  time 
required  for  the  transfer  of  the  energy  from  the  primary  to 
the  secondary  through  the  magnetic  coupling  and  back 
again  was  approximately  equal  to  eight  complete  oscilla- 
tions between  the  magnetic  and  dielectric  fields  of  either  the 
primary  or  the  secondary  circuits.  The  oscillations 
decrease  in  magnitude  due  to  the  Ri2  losses  and  practically 
all  of  the  energy  was  dissipated  into  heat  in  ^0  of  a  second. 

For  the  oscillogram  in  Fig.  134  the  primary  circuit  was 
opened  at  the  instant  all  the  energy  had  been  transferred 
from  the  primary  to  the  secondary  circuit,  thus  preventing 
its  return  to  the  primary  circuit.  Hence  the  secondary 
continues  to  oscillate  until  all  the  energy  has  been  dissi- 
pated as  heat  by  the  Ri2  losses. 

The  oscillogram  in  Fig.  135  shows  the  starting  oscillatory 
transient  of  two  inductively  coupled  circuits  when  an 
alternating  current  of  resonance  frequency  is  impressed  on 
the  primary.  Similar  oscillograms  showing  the  oscillatory 
transfer  of  energy  between  the  primary  and  secondary  of 
dielectrically  coupled  circuits  are  shown  in  Figs.  136,  137 
and  138.  The  difference  in  form  in  the  three  oscillograms 
is  due  to  change  in  the  degree  of  coupling  as  indicated  by 
the  quantitative  data  in  each  case. 

Coupling  Coefficient. — In  coupled  circuits  as  in  Figs. 
130  and  131,  the  interaction  will  depend  on  what  part  of 
the  total  magnetic  flux  interlinks  both  circuits.  The  degree 
of  coupling  which  is  often  termed  "loose"  or  " close, " 
depending  on  whether  a  small  or  large  fraction  of  the  flux 
interlinks  both  circuits,  is  quantitatively  expressed  [by 


158 


ELECTRIC  TRANSIENTS 


RESONANCE 


159 


160 


ELECTRIC  TRANSIENTS 


the  coupling  coefficient.  This  is  defined  as  the  ratio  of 
the  mutual  reactance  to  the  square  root  of  the  product  of 
the  primary  and  secondary  circuit  reactances. 


FIG.   135. — Transient  oscillations.     Inductive  or  magnetic  coupling.     Resonant 

charge. 

Impressed  frequency  =  750  cycles;  R  =  6.5  ohms;  L  =  0.205  henrys;  C  = 
0.2  microfarads;  coefficient  of  coupling  =  11  percent;  timing  wave  100  cycles; 
natural  frequency  790  cycles  when  K  =  0. 


Inductive  coupling  coefficient,  Fig.  131: 
am  M 


(300) 


M  =  mutual  inductance 

L^  =  inductance  of  primary  with  the  secondary  open 

or  removed 
L2  =  inductance  of  secondary  with  the  primary  open 

or  removed. 
Condensive  coupling  coefficient,  Fig.  132: 


cX, 


vc 
c, 


C1  I 

=  V(C. 


c  r 

\j  a\*/ 


(C.  +  C.)  (C.  +  C 
Cm  =  condensance  in  common  condenser 


-,     (301) 


RESONANCE 


161 


, 

II 


0> 

I! 

o3    o 


11 


162 


ELECTRIC  TRANSIENTS 


O 

£d 


I 

O 


S  b 


I" 


II 

g  3 

2  2 
fl 


RESONANCE 


163 


s§ 
-i* 


"Eg 


s 

-^   o 


i  s 

§^ 


1 64  ELECTRIC  TRANSIENTS 

Ca  =  condensance  in  primary  circuit 
Ci  =  condensance  in  secondary  circuit 

C  C 

d  =  -^j—  ~mr-  =  total  condensance  in  primary 

C  o    ~\~    Cm 

C  C 
Cz  =  „     ~/V    =  total  condensance  in  secondary. 

Cb    ~\-    (jm 

Multiplex  Resonance. — In  complex  circuits  or  series  of 
double  energy  loops  the  conditions  for  resonance  may  be 
satisfied  for  more  than  one  frequency  of  the  impressed 
voltage.  The  degrees  of  freedom,  or  the  number  of  fre- 
quencies at  which  resonance  may  occur,  depends  on  the 
number  and  interconnection  of  the  elemental  double 
energy  circuits  in  the  system.  Thus,  a  transmission  line 
having  uniformly  distributed  R,  L,  G  and  C,  and  hence  to 
be  considered  as  consisting  of  an  infinite  series  of  infinitesi- 
mal double  energy  circuits,  would  resonate  for  the  funda- 
mental frequency  of  the  line  as  a  unit  and  for  any  multiple 
or  harmonic  of  the  fundamental  frequency.  As  the  line 
constants  are  not  perfectly  constant  and  the  distribution 
of  R,  L,  G  and  C  not  quite  uniform,  resonance  is  limited  to 
the  fundamental  and  a  few  of  the  lower  harmonics. 

Resonance  Growth  and  Decay. — As  stated  in  the  begin- 
ning of  this  chapter  resonance  in  electric  circuits  implies  a 
forced  oscillation  of  energy  between  magnetic  and  dielectric 
fields,  at  such  frequencies  of  the  impressed  voltage  as  to  make 
the  total  impedance  or  admittance  a  minimum.  To  supply 
the  resonating  circuit  with  the  oscillatory  energy  necessitates 
a  transient  starting  period  during  which  the  amplitude  of 
each  oscillation  is  greater  than  the  one  preceding.  For 
systems  having  constant  finite  circuit  constants  in  which 
the  resonance  phenomena  reach  permanent  values,  the 
growth  of  the  transient  follows  the  exponential  law.  This 
increase  in  the  magnitude  of  the  oscillations  during  the 
starting  period  is  illustrated  by  the  oscillograms  in  Figs. 
139  and  140.  In  these  oscillograms  the  power  supply  was 
cut  off  when  the  resonance  had  reached  the  permanent 
stage.  The  decay  parts  of  the  oscillograms  in  Figs.  139  and 


RESONANCE 


165 


B^ 


03     O) 

a  a 


a  T2 

c3    to 


I 


166 


ELECTRIC  TRANSIENTS 


FIG.   140. — Resonance  in  high  speed  signaling. 

R  =  10  ohms;  L  =  89  millihenrys;  C  =  0.25   microfarads;  timing  wave  100 
cycles;  frequency  =  1070  cycles;  decrement  =  0.052. 


FIG.   141. — Resonance  limited  by  spark  gap  discharge. 

R  =  15  ohms;  L  =  89   millihenrys;  C  =  0.25  microfarads;  timing  wave  100 
cycles;  frequency  =  1070  cycles;  decrement  =  0.079. 


RESONANCE  167 

140,  represent,  therefore,  free  oscillations  with  a  decrease  in 
amplitude  as  the  electric  energy  is  dissipated  into  heat. 

In  Fig.  141  the  starting  period  is  of  the  same  form  as  in 
Fig.  139  or  140,  but  not  the  decay  stage.  It  is  evident  from 
the  circuit  connections  that  the  decay  of  the  resonating 
currents  or  voltages  will  differ  in  shape  depending  at  what 
instant  in  the  cycle  the  short  circuit  occurs.  The  oscillo- 
gram  in  Fig.  141,  for  which  the  short  circuit  was  produced 
by  spark-over,  occurred  near  the  maximum  point  of  the 
voltage  wave  with  practically  all  of  the  oscillating  energy 
initially  stored  in  the  dielectric  field  of  the  condenser. 

Problems  and  Experiments 

1.  Take  oscillograms  showing  the  transients  accompanying  the  growth 
and  decay  of  cumulative  resonance  in  circuits  similar  to  Figs.   139,   140 
and  141. 

2.  Take  oscillograms   of   the   transient  oscillations   of   two   inductively 
coupled  circuits  similar  to  Figs.  133,  134  and  135. 

3.  Take  oscillograms  of  the  transient  oscillations  in  two  dielectrically 
coupled  circuits  similar  to  Figs.  136,  137  and  138. 


CHAPTER  IX 
OSCILLOGRAMS 

In  the  preceding  chapters  the  fundamental  principles 
of  electric  transient  phenomena  are  illustrated  by  a  number 
of  oscillograms,  many  of  which  the  student  should  repro- 
duce in  order  to  gain  the  necessary  appreciation  of  the 
quantitative  value  of  the  factors  involved.  However,  the 
laboratory  work  in  the  course  should  not  be  restricted  to 
the  reproduction  of  oscillograms  appearing  in  the  text  for 
which  quantitative  data  are  provided,  or  to  the  taking  of 
other  oscillograms  that  merely  illustrate  the  fundamental 
principles.  For  while  the  gaining  of  clear  concepts  of  the 
basic  laws  of  transient  electric  phenomena  is  of  primary 
importance,  training  in  applying  the  principles  to  practical 
engineering  problems  is  likewise  an  essential  part  of  the 
work.  Ample  material  for  this  purpose  is  available  in  all 
electrical  engineering  laboratories.  The  oscillograms  in  this 
chapter,  Figs.  142  to  161,  which  were  selected  from  the  labo- 
ratory reports  of  students  in  the  introductory  course  in 
electric  transients,  may  be  taken  as  typical  examples.  The 
students  were  required  to  outline  the  problem,  to  select  the 
necessary  apparatus  and  instruments,  to  make  preliminary 
calculations  and  to  predict  the  form  and  shape  of  the 
transients  to  be  recorded.  They  made  all  the  adjustments 
on  the  oscillograph,  obtained  experimentally  the  recorded 
quantitative  data,  took  the  oscillograms,  developed  the 
films  and  prepared  a  report  on  the  transients  photographic- 
ally recorded  by  the  oscillograph.  Each  oscillogram  repre- 
sents a  separate  problem  to  be  analyzed  on  the  basis  of  the 
principles  discussed  in  the  preceding  chapters. 

168 


OSCILLOGRAMS 


169 


. 

2  a 


. 

i^  ° 


170 


ELECTRIC  TRANSIENTS 


C    « 
O    > 


CSCILLCGRAMS 


171 


172 


ELECTRIC  -TRANSIENTS 


FIG.   145. — T.  A.  regulator  operating  transients. 

Fi  =  exciter  field    current;    V*  =  alternator   field    current;    Va  =    alternator 
terminals. 


FIG.   146. — Undamped  oscillograph  vibrator  oscillations. 

Vi  =  timing  wave,  100  cycles;  Vz  =  Oscillations  of  undamped  oscillograph 
vibrator  superimposed  on  tungsten  lamp  starting  transient.  Vz  =  starting 
transient  (vibrator  damped)  of  tungsten  lamp,  imperfect  contact. 


OSCILLOGRAMS 


173 


174 


ELECTRIC  TRANSIENTS 


OSCILLOGRAMS 


175 


"C  o 

c  o 


§  ^ 


1? 


II 

11 1 

«    03 


:§! 


is, 


s 


176 


ELECTRIC  TRANSIENTS 


FIG.   150. — Current  transformer  transients. 

Vi  =  secondary  current;  Vz  =  secondary  voltages;  Va  =  primary  current; 
primary  /  =  60  amps.;  secondary  /  =  3.5  amps.;  core  undersaturated  before 
transient. 


FIG.  151. — Single  phase  short  circuit  on  a  two-phase  alternator. 
Open  phase  voltage  =  605  volts;  short  circuit  current  =  23  amps.;  E,  field  = 
500  volts;  I,  field  =  3.25  amps.;  frequency  =  60  cycles;  Vi  =  open  phase  voltage; 
Vz  =  short  circuit  current;  V»  =  field  current;  brushes  sparking. 


OSCILLOGRAMS 


477 


te  -j 
^  s 


J  o 

ll 


>>  „ 

Cj    H^ 


-+ 

8  g 


« a 

O    03 


12 


178 


ELECTRIC  TRANSIENTS 


ii 

CD    S 


03    o 

-as 


£••= 
*  a 


C     3 

•Ss 
§3 

•Si 


o   & 

^  s 


OSCILLOGRAMS 


179 


•^.-a 
is  5 


5  II 


fl  T3 
O    (3 


an 
C3 
O  <N 


il 


180 


ELECTRIC  TRANSIENTS 


It- 


So 


OSCILLOGRAMS 


181 


3c 


182 


ELECTRIC  TRANSIENTS 


ll 


o 


O    (H 


S    g 


OSCILLOGRAMS 


183 


tif 

o  g 


H 

is  a 


184 


ELECTRIC  TRANSIENTS 


OSCILLOGRAMS 


185 


N 


•35 

O    o 

•ga 


o  ft 

•8  a 
a  * 


186 


ELECTRIC  TRANSIENTS 


12 


OSCILLOGRAMS  187 

Problems  and  Experiments 

1.  Take  oscillograms  of  a  number  of  transients  in  circuits  of  the  types 
shown  in  this  chapter.     In  each  case  obtain  quantitative  data  and  pre- 
pare'a  report  giving  an  explanation  of  the  transients  appearing  in  the  oscillo- 
gram  based  on  the  fundamental  principles  of  transient  electric  phenomena. 

2.  Find   several    electric    transients    in    the   laboratory   under    different 
circuit  conditions  from  those  described  in  the  book.     For  each  case  draw 
diagrams  of  the  proposed  circuit  connections  showing  the  location  of  the 
vibrators;  make  preliminary  calculations  as  to  the  amount  of  resistance 
required  in  each  vibrator  circuit;  the  most  desirable  speed  of  the  film  drum, 
etc.,  to  give  a  well  proportioned  oscillogram;  take  the  oscillogram;  record 
the  quantitative,  data;  develop  the  film  and  make  prints.     Compare  the 
predicted  forms  of  the  curves  with  the  photographic  record  and  check  the 
preliminary  calculations  with  the  final  circuit  data.     Prepare  a  report  on 
the  transients  recorded  on  the  oscillogram. 


APPENDIX 

Developing  and  Printing  Oscillograms. — The  finished 
oscillogram,  even  if  perfect  electrically,  is  often  disappoint- 
ing photographically.  Care  and  cleanliness  in  the  manipu- 
lation of  the  photographic  film  and  printing  paper  will  reduce 
these  failures  to  a  negligible  quantity. 

Starting  with  the  unexposed  film,  the  photographic  proc- 
ess will  be  traced  to  the  completed  print,  ready  for  the  files. 
Cleanliness  is  essential.  During  no  part  of  the  process 
should  the  hands  come  in  contact  with  the  sensitized  side 
of  the  negative.  In  order  to  accomplish  this,  the  film  and 
its  black  protecting  paper  should  be  placed  on  the  drum  as 
a  unit,  with  the  black  paper  on  the  outside.  After  the  film 
and  paper  have  been  adjusted  to  the  proper  position,  the 
paper  may  be  removed  from  the  drum.  In  this  way  the 
hands  have  not  touched  the  surface  of  the  film. 

Unlike  most  photographic  work,  the  permissible  time  of 
exposure  for  oscillograms  is  limited,  especially  in  high  speed 
work.  Stray  light  of  any  nature  is  injurious.  For  this 
reason  it  is  highly  desirable  to  load  the  film-holders  in 
complete  darkness  and  to  develop  for  the  first  two  or  three 
minutes  without  even  the  ruby  light.  After  a  little  practice 
the  student  will  have  no  trouble  in  working  without  the 
darkroom  light. 

Any  metol-hydrochinon  film  developer  may  be  used  with 
varying  degrees  of  success.  Where  only  a  fewT  negatives 
are  made  at  odd  times,  Eastman's  " Special"  developer  is 
satisfactory.  This  developer  will  give  better  results  if 
some  of  the  used  developer  be  added  to  the  fresh  solution. 
In  our  laboratories  the  following  stock  solution  is  used: 
water  64  oz.,  metol  one  drachm,  hydrochinon  one-half  oz., 
sodium  sulphite  2  oz.,  sodium  carbonate  3  oz.,  potassium 
bromide  30  grains.  This  stock  solution  is  diluted  in  the 
proportion  of  two  parts  stock  solution  to  one  part  water. 

188 


APPENDIX  189 

It  is  very  important  that  the  developer  be  used  at  a  tem- 
perature of  65  deg.  F.  The  hydrochinon  is  inactive  at  lower 
temperatures,  resulting  in  slow  development  and  a  flat 
negative  which  lacks  density  and  contrast.  If  used  at  a 
higher  temperature,  the  negative  will  gain  density  rapidly 
but  will  be  lacking  in  contrast  and  show  a  decided  tendency 
to  fog  in  the  unexposed  portions. 

The  exposed  negative  should  be  given  maximum  develop- 
ment possible  without  fogging  the  unexposed  portions. 
The  image  should  be  allowed  to  develop  until  it  appears 
quite  definite  on  the  reverse  side  of  the  negative.  A  good 
rule  to  follow  is  to  develop  until  by  comparison  with  the 
back  of  the  negative,  the  sensitized  side  appears  quite 
gray.  The  gray  tone  will  disappear  in  the  fixing  bath  and 
further  development  is  detrimental. 

Care  should  be  taken  to  fix  and  wash  the  negatives  prop- 
erly. The  film  should  be  left  in  the  standard  fixing  bath 
at  least  five  minutes  longer  than  is  necessary  to  dissolve  the 
last  visible  trace  of  un-reduced  silver  salts.  After  careful 
fixing,  the  film  should  be  washed  for  at  least  twenty  minutes 
in  running  water.  It  is  desirable  to  rinse  off  the  surface 
with  a  tuft  of  cotton  before  hanging  up  to  dry.  The  hurry 
which  often  comes  in  the  completion  of  the  day's  work  in 
the  laboratory,  results  in  haste  in  the  darkroom.  If  the 
fixing  and  washing  processes  are  slighted,  the  film,  though 
apparently  good  at  the  time,  becomes  worthless  in  a  few 
months  on  account  of  staining. 

The  same  developer  may  be  used  for  the  printing  paper, 
except  that  it  should  be  always  mixed  fresh  just  before 
using.  The  best  results  are  obtained  by  following  the 
printed  instructions  accompanying  the  photographic  paper. 

In  order  to  get  the  maximum  contrast  in  the  finished 
print,  it  is  necessary  to  use  the  most  contrasting  photo- 
graphic paper.  The  paper  which  has  proven  the  best  is 
the  Eastman  "Azo,"  grade  No.  4,  glossy,  although  others 
may  satisfy  the  individual  user.  If  this  is  purchased  in  ten 
yard  rolls,  twenty  inches  wide  and  cut  on  a  circular  saw  or 


190  APPENDIX 

band-saw  to  four  and  one-half  inch  widths,  four  small  rolls 
result  with  a  two  inch  roll  left  over  for  use  in  testing 
exposure. 

Prints  should  be  given  normal  exposure  so  that  with 
normal  or  full  development  the  background  reaches  good 
density  without  appreciable  reduction  of  the  silver  in  the 
highlights.  As  usual,  prints  should  be  fixed  fifteen  minutes 
in  a  standard  hypo  bath  and  washed  for  at  least  twenty-five 
minutes  in  running  water.  The  best  finish  is  obtained  by 
drying  the  prints  on  ferro-type  plates,  which  imparts  high 
gloss  to  the  surface. 


INDEX 


Alternating  current  transients,  40 
Alternator  field  transients,  50,  61 
Alternators,  single-phase,  61-69 

three-phase,  53-60,  69 

two-phase,  71,  176 
Arcing  grounds,  141 
Armature  reactance,  59 

reaction,  58 

transients,  50,  61 
Artificial  electric  lines,  101 
Asymmetrical  field  transients,  63 
Attenuation  constant,  30 

B 

Breaking  field  circuit,  138 
"Bucking  broncho,"  175 


Capacitance,  7,  9 
Carbon  lamps,  132 
Circuit  breakers,  170 

constants,  101,  130 
Compound  circuits,  126 
Condensance,  5,  7,  9 

variable,  142 
Condensive  coupling,  156 
Conductance,  9 
Corona,  143 
Coulombs,  9 
Coupled  circuits,  154 
Coupling  coefficient,  155 
Current  resonance,  148 

D 

Damping  factor,  92,  115 
Developing  oscillograms,  188 
Dielectric  circuit,  4 
coupling,  156 


Dielectric  field  intensity,  9 

flux,  4,  9 

gradient,  9 
Direct  coupling,  156 

current  transients,  24 
Dissipation  constant,  30,  92,  115 
Distance  angle,  116 
Double  energy  transients,  21.  75-100 


Elastance,  4,  9 
Electric  circuit,  7 

line  oscillations,  101 
Energy,  8,  9 
Exponential  curve,  32 

law,  24 


Farad,  7,  9 

Faraday's  lines  of  force,  2 
Forming  magnetic  field,  27,  139 
Frequency,    distributed    R,    L,    G 

and  C,  110 

massed  R,  L,  G  and  C,  111 
Frohlich's  formula,  135 


Galvanometers,  14 
Generator  field  transient,  138 
Gilbert,  9 


II 


Henry,  4,  9 

High  frequency  signalling,  166 


Impedance,  9 
Impulses,  124 


191 


192 


INDEX . 


Inductance,  3,  9 

variable,  135 

Induction  motors,  20,  48,  177,  181 
Initial  values,  33 
Intermittent  arcs,  141 


Joule,  8,  9 


Permittivity,  4,  6 
Phase  angle,  103 
Polyphase  short  circuit,  50 
Power  surges,  114 

transfer  factor,  123 
Printing  oscillograms,  189 
Pulsating  condensance,  142 

inductance,  135 

resistance,  133 


K 
Kirchoff's  Laws,  79,  83 

L 

Leaky  condenser,  83,  104 
Length  of  line,  112 
Lifting  magnet  transient,  169 
Line  constants,  101 
oscillations,  101 
Lumpy  line,  101 


R 


Reactance  curves,  149,  154 
Reluctance,  3,  9 
Repulsion  induction  motor,  178 
Resonance,  145 
"Resonance"  frequency,  78 
growth  and  decay,  164 
Resistance,  9,  130 
Resistivity,  9 

Rotary  converter,  182-187 
Rotating  magnetic  field,  48 


M 

Magnetic  circuit,  2 
coupling,  156 
field  intensity,  3,  9 
flux,  2,  9 

Microfarads,  7 

Multiplex  resonance,  164 

N 

Natural  admittance,  77 
impedance,  77 
period  of  oscillation,  108 


Oersted,  9 
Ohm's  law,  2,  4,  9 
Oscillator  alternator,  17 
Oscillatory  circuits,  79 
Oscillograms,  20 
Oscillographs,  10 


Permeability,  3,  9 
Permittance,  9 


Series  generator,  173 
Sharpness  of  resonance,  147 
Short  circuits,  polyphase,  50 

single-phase,  61 
Single  energy  transients,  a.-c.,  40 

d.-c.,  23 

Space  angles,  103,  116 
Split-phase  motor,  178 
Standing  waves,  118 
Surge  admittance,  77 

impedance,  77,  113 
Susceptance  curves,  154 
Synchronous  reactance,  58 


T.  A.  regulator,  172 
"T"  circuits,  104 
Three-phase  transients,  44 
Time  angles,  103,  116 

constant,  28 
Timing  waves,  17 
"T"  line,  102 
Transformers,  135,  176 


INDEX 

Transmission  line,  artificial,  101  Units,  9 

constants,  101  V 

equations,  114 

oscillations,  101  Variable  circuit  constants,  130 

Traveling  waves,  116  Velocity  unit  of  length,  113 

Tungsten  lamp,  1'31  Voltage  resonance,  145 

U  W 

Undamped  vibrator,  172  Watt,  8 


193 


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